Integral Equations and Operator Theory, Vol, 3/2 1980 © Birkhfluser Verlag, CH-4010 Basel (Switzerland), 1980

NONNORMAL DILATIONS, DISCONJUGACY AND CONSTRAINED

SPECTRAL FACTORIZATION

Joseph A. Ball and J. William Helton*

Introduction.

A basic technique in studying an operator T on a Hilbert space H is to

construct a Hilbert space H containing H and a simple operator M on H so

that

(I.1) Tn = PH Mnl H .

Such dilations with unitary [N-F] (or normal) M have been heavily used and

the structure arises in physical situations. The next natural step is to study dila-

tions which are non-normal. This paper treats an extremely simple (probably

the simplest) situation of that type and the surprising conclusion is that the

structure is deep (indeed has much more depth than the structure for dilation

to a unitary operator), and is closely connected with classical differential equa-

tions. In fact this paper might interest researchers in ordinary differential equa-

tions, because what we do here gives a radical new approach to Sturm-Liouville

disconjugacy theory (and yields very general theorems).

*Research on this paper was supported by NSF grant nos. MCS77-00966 and MCS77-01517.

Ball and Helton 217

The dilations we study are based on an operator M which is Jordan,

namely, one of the form M = S + N where S is self-adjoint and N is a nil-

potent operator which commutes with S. When N 2 = 0, we say that M is

Jordan of order 2. An argument of D. Sarason [S] shows that for T to have a

Jordan dilation M as in (I.1) the space H must be the direct difference of 2

subspaces H1 and H2 of H which are invariant under M. This sets the para-

digm for studying dilation problems: One must give a simple characterization

of

(I.2) Jordan operators.

(I.3) Sub-Jordan operators--the restriction of a Jordan operator to an

invariant subspace.

(I.4) Compressed Jordan operators--an operator of the form (I.1)

with M Jordan.

Our study is based on analyzing the function

Or(s) = e -isT* e is'r

which we call the symbol expansion of T. If Or is an operator polynomial

n

Or(s) = ]~ skBk, k = l

we call it a coadjoint operator of order n. Throughout the paper we restrict atten-

tion to order 2 operators. The answer to problem (I.2) is:

THEOREM. [HI] T is a Jordan operator (order 2) if and only if T and T*

are coadjoint (order 2).

The likely solution to problem (I.3) is

Conjecture. T is sub-Jordan of order 2 if and only if T is coadjoint.

Ball and Helton 218

This paper settles a large piece of the conjecture affirmatively.

Our technique is based on spectral factorization of the symbol expansion

QT. Since QT is a polynomial it has both an outer and ,-outer factorization (in

the sense of [R-R]) and by Theorem [R-R], these are first order polynomials.

Thus we have

(I.5) QT(S) = (V+sA)*(V+sA)

(I.5") QT*(S) = (V+sA,)*(V,+sA,)

Immediately the factorizations give some integers which are unitary invariants

for a coadjoint operator T. Define the defect of T to be the pair of integers

Def(T) = (dim corange V, dim corange A)

Def, (T) = (dim corange V,, dim corange A)

Unfortunately, our analysis of coadjoint operators is complete only when the

defect is (0,0). However, we are of the strong opinion that the methods

herein form a very solid basis for handling the general case (large parts of the

paper never use the defect --(0,0) hypothesis). Our main theorem is:

THEOREM I. If T is a coadjoint operator with either defect = (0,0) or *-

defect = (0,0), then T has a Jordan extension.

Some examples illustrate the connection of this theorem with classical

mathematics and provide a gauge of its power. The basic example of a coad-

joint operator (order 2) is T = Mh multiplication by a C I(R) function h on

the space of Hilbert space -0-valued absolutely continuous functions with inner

product

I

(I.6) (f,g> = _f {(Pf',g')n + (Qf,g)n} dx . -0

Ball and Helton 219

Here P and Q are self-adjoint operator-valued functions with P >/~ I. And

we denote the space of all such functions with finite norm by S. Any such

sesquilinear form (-,-> on a domain S gives rise to a positive definite operator

L on the L 2 space of -0-valued functions such that DomL '/-~ = Sand ( f , g ) = 1

f0 (Lf,g)ndx for f in DomL (cf., [D-S]). In the case here L is the Sturm-

Liouville operator

d p d (I,7) Lf - - - d---~ T x + Q

with free boundary conditions. Various subspaces of S correspond to various

other boundary conditions associated with L. For example, S0o =A_

{f6 S: f(0) = f(1) = 0} corresponds to considering L on the domain D of

functions which vanish at the ends while S01 --- {f 6 S: f(0) = 0} corresponds

to the domain of f with f(0) = 0 = f'(1). (In this notation S corresponds to

Sll.) As we shall see work with these examples indicate that the defect of

T = Mn is closely related to choice of boundary conditions. The examples

strongly suggest

CONJECTURE. The defect or ,-defect of T = Mh restricted to Sa,b is (0,0)

if and only if the associated boundary conditions at 0 are adjoint to those at 1.

("Adjoint" will be defined precisely in §3.)

Theorem I is closely related to a classical theorem about Sturm-Liouviile

operators, a special case of which is:

THEOREM. (cf., [R], Proof of Theorem IV.7.4). If the Hermitian form (I.6)

is positive- definite on S00, then it has a representation as

Ball and Helton 220

where /3 is some self-adjoint operator valued function on [0,1]. Hence, the

associated Sturm-Liouville operator L has a factorization

This will look very familiar to those who know classical conjugate point theory

(typically /3 = - u'/u where u is a solution to Lu = 0 which never vanishes

on [0,1]); also /3 is a solution of the Riccati equation associated with L.

This paper gives a procedure which should extend this type of result

greatly, namely, to differential equations with operator coefficients which are

highly discontinuous or even pathological. This might give new ways to obtain

solutions u to intractable equations. These things would be accomplished by

removal of the defect = (0,0) hypothesis: at present we obtain these results

but under an unacceptable hypothesis.

J. Agler [A] has recently announced considerable progress on the first

conjecture above. His technique is to approximate an arbitrary coadjoint opera-

tor with a "smooth" one for which the conjecture is already known to be true.

He uses Arveson's theory of extensions of completely positive maps in a very

interesting and structurally significant way to analyze convergence of the

"smooth" operators. From this he is able to prove that a cyclic coadjoint opera-

tor (order 2) has a Jordan extension. His technique also appears to give infor-

mation about the class of compressed Jordan operators ((I.4) above).

There are additional topics in the paper. The example we just gave of a

coadjoint operator is actually close to the general case. In §3 it is shown that

every sub-Jordan operator has a representation as a multiplication operator on a

general type of Sobolev space.

In §4 we treat the finite dimensional case and prove

Ball and Helton 221

THEOREM II. Every finite dimensional coadjoint operator is Jordan.

Also we give equations connected with problem (I.4). In §5 we give an exam-

ple which shows that simultaneously lifting n-commuting coadjoint operators

commuting in a strong sense corresponds to classical disconjugacy theory for

partial differential equations on R n. In §6 we discuss the operator Mz of mul-

tiplication by z on a Sobolev space of analytic functions. We show that such

an operator is a special case of a more general type of operator recently dis-

cussed by Cowan and Douglas [C-D].

We conclude the introduction with a bit of perspective and speculation.

One can look at lifting theory very algebraically. To wit the Nagy lifting

theorem says that the relationship

T*T~< I lifts to U * U - - - U U * = I .

The commutant lifting theorem says

Tj*Tj ~< I and TtT2 = T2TI

U 3Uj=UjUj = I

liftsto

and UIU2 = U2UI.

This paper concerns extending

T . 3 - 3T*2T+ 3 T * T 2 - T 3= 0 to

j , 3 3 j , 2 j + 3 j , j 2 _ j 3 = 0

j 3 - 3 j 2 J * + 3 j j * 2 - j * 3 = 0 .

and

The surprising thing is that these very algebraic considerations contain an enor-

mous amount of classical analysis. The commutant lifting theorem is an

extremely strong, very concrete generalization of classical Nevanlinna-Pick

Caratheodory-Fejer-Schur-etc. interpolation theory (with considerable

Ball and Helton 222

engineering application). As we see here Jordan extensions are intimately tied

to differential equations. All of this certainly makes one wonder which other

algebraic relationships lift and how much classical analysis as well as other

mathematics can be subsumed by the lifting of operators.

TABLE OF CONTENTS

Introduction

PART I. Jordan Extensions of a Single Coadjoint Operator

§I. Jordan Extensions

§2. Spectral Factorization

PART II. Examples

§3. Differential Operators

§4. Compressed Jordan Operators

PART lII. Several Coadjoint Operators

§5. Commuting Jordan and Coadjoint Operators

§6. Complex Jordan Operators

Ball and Helton 223

PART I. E X T E N D I N G A SINGLE COADJOINT O P E R A T O R

§1. Jordan Extensions.

The basic problem is to construct a Jordan extension J = S q-N on

K D H for a given coadjoint T on H. It turns out that the key to building J

lies in finding the operator PHN]H-- A. In this section we show that such an

A satisfies a set of three operator equations and one operator inequality whose

coefficients are determined by T. Then we prove that any A satisfying these

equations gives a Jordan extension J of T and in fact determines it uniquely

(provided J is minimal). The remarkable thing is that these equations can

frequently be solved, and in fact read off directly from the outer spectra! factor-

ization of Qx(s). Solving the equations is the subject of §2. Note that this sec-

tion never uses the defect (0,0) hypothesis; the results in this section are

necessary and sufficient and quite complete. We begin the section by stating

the main result, then we give motivation and then the proof.

We give the basic equations which A = PHN[H must satisfy as a

definition.

DEFINITION 1.1. Given a coadjoint operator T on H with symbol expan-

sion

QT(s) = I + sB l + s2B2

the operator A on H is said to be a nilpart for T if

(1.1) A*A ~< B2

Ball and Helton 224

(1.2) A - A * = T - T* = - i B l

and

(1.3) T*(A + A*) = (A + A*)T.

The operator A is said to be a strong nilpart for T if in addition

(1.4) Bzf, f ~< k{]lfl[ 2 -]lWf]~} (for some k < oo) ,

where W is the unique contraction operator such that BS-~W = A* and

Ran W c [Ran B)/:] -.

Note that W is a contraction because of (1.1) (see [D]). We shall see

that the existence of anilpart for T allows us to construct formally a possibly

unbounded Jordan extension J. The inequality (1.4) insures that J is

bounded. The main theorem of this section is

T H E O R E M 1.1. If T on H has a Jordan extension S + N on K then

A -- P t tNIHis a strong nilpart for T. Conversely if T is coadjoint and A is a

strong nilpart for T, then T has a Jordan extension S + N with A -- PHNIH.

Throughout we assume all orders equal 2.

Note that if we write

T - T * A = U + - -

2

then (1.1) and (1.2) become

B2 ,

and

Ball and Helton 225

(1.2') U = U*.

An operator equation of the type (1.1') with equality is called an algebraic

Riccati equation; such equations form the cornerstone of classical optimal con-

trol theory, and as such are well understood. The type appearing here is the

simplest kind because the coefficient of U 2 is invertible. What we have here

(for the equality case) is an algebraic Riccati equation with the linear side con-

dition (1.3). The linear side condition (1.3) makes the problem much harder;

however we can handle it in many cases.

There are several reasonable approaches to proving Theorem 1.1. One is

to represent K as H ~ H and then construct a nilpotent N on K with

A = PHNIH. The nilpart equations imply the construction works. Next one

shows that a self-adjoint S with [ S , N ] = 0 exists on K so that J = S + N

has H for an invariant subspace. This was the first approach we took and it

was carried through under more stringent hypotheses. The second method fol-

lows the differential equations approach closely and that is the one we present.

Since it is very formal and algebraic, it behooves us to start with motivation.

The differential equations example which we now present extends the one in

[H1] [H2] to more general boundary conditions.

Let L be a positive-definite Sturm-Liouville operator

d 2

L = dx 2 + q

(q continuous real-valued on [0,1]) with

f 6 AC[0,1], f '6AC[0,1], f"6L2(0,1),

a~ is a real constant}. Then the inner-product

I

(f,g>H = fo (Lf)~dx

domain D = {f 6 L2(0,1):

alf(1) + f'(1) = 0 where

Ball and Helton 226

is positive definite on D. It follows via an integration by parts that the comple-

tion of D in this inner product can be identified as

H = {f ~ L2(0,1): f 6 AC[0,1], f' 6 L2(0,1)}

where

<f ,g>H : fqf~dx + ff '~ 'dx + alf(1) g(1).

The coadjoint operator T of interest here is multiplication by x on H.

The following criterion for positive definiteness of L is basic in sufficiency

theory of the calculus of variations.

THEOREM. (cf. [G-F]) L is positive-definite if and only if the solution u of

- u " + qu = 0 satisfying u(0) = 1, u'(0) = 0 also has u(x) > 0 for

0~< x~< 1 andu ' (1) + a l u ( 1 ) > 0 .

The theorem in [G-F] is stated for f(0) = 0 = f(1) boundary conditions

but the proof easily adapts to handle the boundary conditions considered here.

Theorem IV.7.2 in Reid's book [R] gives a matrix generalization of this result.

d 2 We apply the theorem not to L = dx 2 + q i t s e l f b u t to L~ L 8I

for a small number 8 > 0 with the property that L >~ 8I.

because of the compactness of the resolvent of L. Let u

L8 u = 0 obtained through the theorem and set/3 = - u'/u.

/3 is differentiable on [0,1]

Such a 8 exists

be a solution to

Since u(x) > 0,

and /3 satisfies the Riccati equation

f l '+q - -8 =/32.

It then follows for all f

1 1

Ilfll~ -- fo (If'12 + (q-a)lfl2)dx + 8 fo Ifl2dx + atlf(1)12

Ball and Helton 227

1 1 1 =folfl2(q-8+,8')dX+fo (]fl2)'/JdX+foJf'J2dx 1

+ aJ'0 Ifl2dx -Ifl2/JI d + a~ If(l)12

= II G3 + d ) Ilfll~2 + Ilfll~l

where K2 = L2(0,1) and Kl = L2(Sdx + (al -/3(1))/zl +/3(0)/z0) with /Xl

and tz0 the point mass measures at 1 and 0 respctively.

To build the Jordan extension of T define K = Kl • K2 and J to be I

multiplication by 1~ ~] on K. Clearly J i s a Jordan operator. The operator

(~ + d-~-)

is an isometry of H into K. Use (xf)' = f + xf' to verify that J i -- iT. Hence

if H is identified with the range of i, then it is invariant under J and T = JIH is exhibited as being sub-Jordan.

There are other relationships satisfied by key operators in this constuction.

d ) . If $2 is multiplication by x on K2, then the operator D = (13 + dx '

H ---- K2 satisfies the equation

where i2

DT = $2 D + i2

is the inclusion map of H into K2. It can also be checked that

<B2f, g)H = <f,g>K 2

Ball and Helton 228

where B 2 is the coefficient of s 2 in the symbol expansion of T. This illus-

trates Lemma 1.3 below which is critical in our general algebraic construction.

Now we turn to the rigorous proof of Theorem 1.1. Let us recall some

general facts about coadjoint operators. Define CT: L (H) - , L (H) by

CT(X) = i (XT - T ' X )

and C-p(X) =CT(Cp-I(X)), (n=2 ,3 .... ). Then QT(S) =

being coadjoint (of order 2) is equivalent to CT3(1) = 0 and

QT(S) = I + Bls + B2 s2

= I + i (T-T*)s + I/2 C2(I)S 2,

In this case,

(1.5)

Also

or

(1.6)

eSCs(I). Hence T

1 B 2 = lira ~- QT(S) >/ 0 and Cs(B2) = I/2 CT3(I) = 0 implies

T* B2 = B2 T .

T'B2 T = lim 1 T* e -isT* eiSTT S~°° 7

-- lira 1 e_isT, eisT s-~ ~ - T*T

1 ~< IITII 2 ~irn 7 Qm(s) = IITII2B2,

T*B2T ~< IITII 2 B2.

From (1.5) and (1.6) we get immediately the following

Ball and Helton 229

LEMMA 1.2. Suppose T is coadjoint on H. Let K2 be the completion of

H (with elements of 0-norm identified with 0) in the inner product

(f,g)K2 = (B2f,g>H,

and let i2: H---" K2 be the inclusion map. Then

operator $2 on K2 (i2T = $2i2) such that IIS211 ~< IITII.

Now suppose T is coadjoint and A is a nilpart for

definition of B2,

Hence

A * T =

by (1.2)

= A* AI +

o r

(1.7) A*T = T 'A* + B 2 .

Proof of Theorem 1.1. Suppose

A = PHNIH. Then

T induces a self-adjoint

T. Then by

by (1.3) and the definition of B2

by (1.2) again,

T = S + N[H is sub-Jordan and

Ball and Helton 230

QT(S) = PH e-iSN'eiSNIH

= IH + is(A--A T) + s2 PHN*NI/¢.

Hence B2 = PHN*NIH. But then

A*A = P H N * P H N I H ~< B2

which is (1.1). Also, from the coefficient of s above, we get (1.2).

lowing computation establishes (1..3):

T*(A+A*) = PH(S+N*) PH(N+N*)IH

= PH(S+N*)(N+N*)IH

= PH(N+N*) (S+N)IH

= PH(N+N*)PH (S+N)IH

= ( A + A * ) T.

We have proved that A is a nilpart for T.

To verify that A is a strong nilpart compute that for f fi H,

<B2 f,f>H = <N* N f ,f)x = IINfll 2

= IlNP(kerN~ fll 2 ~< IINII 2 IlPr~nN'fll 2

~< IINII 2 IlPkerN, fll 2 (since N * : = 0 implies

that ran N* c ker N*)

= IlNII2{tlf[I 2 -lipton N f[121.

The fol-

Ball and Helton 231

Hence (1.4) follows with k = IIN[I 2 if we show

(1.4') IlWfll 2 ~< IIL..~fll 2.

Let Aj = NIH: H---. K, so that B2 = Al* Ai. Let A~ have polar decomposi-

tion A~ = Wl * B~/:, where W~: K ---. H is a partial isometry with initial space

equal to [ran Ai]- and final space equal to [ran B2-']-. Then one can easily

check that W = WllH. Hence, for f 6 H,

[IWfll 2 = IIW~ fll 2 = IIW, Pran Nfll 2 ~< IIEa. N fll 2

and (1.4') follows. This completes the proof of necessity in Theorem 1.1.

Before commencing the proof of the converse, we remark that for any

coadjoint operator T, Ile~STII = 0(Isl) as I s [ - - ~ and hence for any function

F in the Schwartz space Son the real line, we can define

1 f F(S) e isT F(T) = ~ -oo ds

where ~'(s) is the Fourier transform of F (see [D-S]). In the case where F

is the restriction to the real line of a function analytic on a neighborhood of the

spectrum of T, this formula agrees with the Riesz-Dunford functional calculus

for T. In particular, if F is zero on a neighborhood of sp(T), then F(T) = 0.

We now isolate the following.

LEMMA 1.3. Let A be a nilpart for the coadjoint operator T, and let i2, K2

and $2 be as in Lemma 1.2. Let K1 be the completion of H (elements of

0-norm being identified as 0) in the norm

Ilfll~, = IIfll~ - I l W f l l ~ .

Ball and Helton 232

and let i1: H - - . KI bc Lhe inclusion map. Then

(1) there is an operator D: H--" K2 such that

and

(a) <Df, i2g>K 2 = <f, Ag)H

(b) DT = S2D + i2

(c) Ilfll 2 --Ili~fl[~l + l l D f l [ ~ 2

(2) there is a self-adjoint operator $1 on

IIS,II ~< IITII.

Proof. First, observe that the

definition of W and condition 1.1.

We note that

I( f, mg>,91 = I<A* f,g >,l = ]<Wf, B~ / g >Hi

~< IlWfllHllu~/:gll. = IlWfllHili2gll,~2

and hence expression (la) defines a bounded operator D from

(Formally, D is simply B~ -tA*.) For f and g in H,

<DTf, i2g}K 2 = <Tf, Ag)H

= < f,T* Ag>H

=<f , ( AT+B2) g ) , q (by 1.7)

= (D f, i2Tg)t¢ 2 +< i2 f, i2g)/< 2

Ki such that Slil = ilT and

K~-metric is positive-definite from the

H into K2.

= (($2 D + i2) f, i2 g)l< 2

Ball and Helton 233

and (lb) follows.

Also

]IDfI]K2 = sup { I(Df, i2g)K2l: Ili2gl]K2 ~ 1}

= Sup{I(f, Ag)HI: I[B~/"gI[H ~< 11

= sup{l(Wf, h)[: ]]hJJH ~< 1} ~lWfllH.

From this (lc) follows immediately.

To prove (2), we first wish to verify the identity

( I -W*W)T = T*(I-W*W), or

(1.8) T - T* = W * W T - T * W * W .

Let B~ -1 be the (possibly unbounded) generalized inverse of B 2. (Since B2

is positive, B5 -1 can be defined most conveniently by the self-adjoint operator

O,t=O calculus as f(B2), where f(t) = | l / t , t > 0 ") Then B2B~ -1 C B5 -l B 2 - - PRanB 2.

Since A*A ~<B 2 by (1.1), A = APRanB2, A* =Pm, nB2A*, and W = BSV:A *.

Then

W* WT - T* W* W = A B71A * T-T* A BS-IA *

= A B~ -I [B 2 + T* A*] - T* ABSqA * (by (1.7))

= A PRan B 2

Since T*B2=B2T by (1.5), ByIT *

becomes

A + A[PRanaffBTI + BS-1T* PIR~nB÷

+ A BS<T * A* - T* A B71A*.

PRanB 2 = PRanBff B2 I. Hence the above

]A* - T*AB~ -1 A*

Ball and Helton 234

= A + ( A T - T * A ) B7 ~A*

= A - B2 B~ -1 A* by (1.7)

= A - A * = T - T * by (1.2) .

Hence equation (1.8) follows.

By iterating (1.8) and considering the power series expansion of the

exponential function, it follows that

e -isT* ( I - -W'W) e isT = I -- W* W

and hence e isr induces a one-parameter unitary group U(s) on KI

(U(s) il = il eisX). By Stone's Theorem, U(s) = e iSsx for some self-adjoint (at

this point possibly unbounded) operator Sl on K1 such that Slil = i lT. (In

particular, the range of il is contained in the domain of S~ and is invariant

under $1.) For f any function in the Schwartz space S of rapidly decreasing

functions on R (see [Y]), the Fourier inversion formula may be used to

define the functional calculus for the self-adjoint operator S~:

c c

f ( s l ) = ?(¢) e ia#sl - - o o

where

t'(~:) = (2rr) -'/: f e -iex f(x)dx.

To show that IlslH ~< [IT[I, it suflfces to show that, for any f 6 S with suppf f'l

[- IITII, IITII] = ,b, and for any x E H, f(S) i x x = 0. Since eiSSq I = il e iST, it

follows that f(S1) il = il f(T) for any f in S, as noted above, if f has support

outside of the spectrum of T, f(T) =0 and hence f(SI)i 1 = 0 as needed.

Ball and Helton 235

CONCLUSION OF PROOF OF T H E O R E M 1.1. Now assume that A

is a strong nilpart for T. It follows immediately from (1.4) that the operator

F: il f---" i2f (f E H) is well-defined and extends to define a bounded operator

(also called F) from K1 into K2 with Ilrll -< k (k

i; o 1 J -- defines a Jordan operator on K~ ~ K2 (S = $2

as in (1.4)). Hence

[S~ $201 is self-adjoint,

is nilpotent of order 2, and it is easy to check that SN = NS on an

1,, f] appropriate dense set). Further, by Lemma 1.3, the operator i: f ~ [Df]

isometrically embeds H into K1 • K2; we compute

i; 1 ,of, :I sif i [ i fl

(F il + $2 D) (i2 + D T - i2)

l,,~fl = = i T f . l DTf]

Hence, if H is identified with its image via i in K 1 (~ K2, then

and T = JIH. Furthermore,

(PiHNif , ig)iH = (Ni f , ig>Kl~ K 2 = (i2 f, Dg)K 2

J: H-"* H

= (Af, g>H (by definition of D) .

Ball and Helton 236

Hence, when the above identification is made, A = PHNIH. Theorem 1.1 fol-

lows.

The next theorem indicates precisely how strong nilparts for a coadjoint

operator T are in one-to-one correspondence with Jordan extensions of T. A

Jordan extension J = S + N on K of the coadjoint operator T on H is said

to be minimal if the smallest subspace of K containing H and invariant for

both S and N is all of K.

T H E O R E M 1.4. Suppose T on H is sub-Jordan with two minimal Jordan

extensions J~ = S i + N~ on K i Z) H ( i= 1,2). Then Jl is unitarily equivalent

to J2 via a unitary leaving H fixed if and only if the strong nilpart

Aa = PHNl[Hassocia ted with Ji is equal to the strong nilpart A2 = PtcN2[H

associated with J2.

Proof. We first consider the case T = J (and H = K), that is, T is Jor-

dan. Then the assertion of the theorem is that if T = Si + Ni where Si = Si*,

Si Ni = Ni Si and Ni 2 = 0 for i = 1,2, then Nt = N2 (and hence also Sl = $2).

Since in this case QT (s) = QN, (s) = QN 2 (s), it follows that

CT(I) = i ( N , - N ~ ) = i ( N z - N 2 )

and

1/2 CT2(I) = N 1 N1 = N 2 N2 .

That N1 = N2 follows immediately from the following lemma, which appears

implicitly in the proof of Theorem I in [HI].

LEMMA 1.5. A nilpotent operator N is uniquely determined from the two

self-adjoint operators Bl = i (N-N*) and B2 = N*N according to the formula

N ' x = B2Bi-lx i f x 6 RanB 2

0 if x Ran B~

Ball and Helton 237

Proof. Let x = B2y E RanB{. Then N*x = N*B12y

N*(NN* + N*N)y = N*NN*y while

i B 2 BFlx = iN*N BI Y = - N*N(N-N*)y

= N*NN*y .

For x E (RanB~) = KerB~ , 0 - - B~x = (NN* + N*N)x and hence

NN* x = - N* Nx E RanN N RanN*, But, since N is nilpotent,

RanN C K e r N = (RanN*) , so we must have N N * x = N * N x = 0 , and

hence N ' x = 0 . Since RanB~+ (RanB~) is dense, we see that Bl and B2

uniquely determine the nilpotent operator N*, and hence N. The lemma fol-

lows.

Proof of Theorem 1.4. (general case): Suppose that T -- Si + N~lHwhere

Si and N i are operators on Ki such that Si *= Si, SiNi = NiSi, Ni 2 = 0 (for

i = l , 2 ) , and suppose A = P H N I I H =PHN2]/-/. Let M i = H + N i H ( i = I , 2 ) .

Then, since Ni is nilpotent, Mi is invariant under Ni; also, for h,k E H,

Si(h + Nik) = [(Si + Ni) - Ni](h + Nik)

= Th - Ni h + NiTk E Mi.

Hence Mi is invariant under both Si and N~ and M~ 2)H. Hence, if

J i = S i + N i on K~ is a minimal Jordan extension of T, M i = K i . For

h ,kE H,

IIh + N, kll,~ = Ilhll 2 + (h,Ak)H + ( A k , h ) g

+ ('/2 C2(I)k,k)H

= l l h + N2kll,~2 •

Ball and Helton 238

Hence the formula

U: h + Nlk ---. h + N2k

defines a unitary operator mapping K1 onto K2 which leaves H fixed.

easy to check that

U(S1 + N0(h + Nlk) = U(Th + N1Tk)

= T h + N 2 T k = (S2+N2)(h+N 2k)

= ($2 + N2)U(h + N1 k) ,

that is

It is

U(S1 + N1) = (S2 + N2)U.

Sufficiency in Theorem 1.4 follows.

Conversely, suppose that T = Si + Ni[H, Si, Ni on Ki as above, and that

there is a unitary map U : K ~ K 2 such that U ( S I + N 1 ) = (S2+N2)U and

U h = h for h 6 H. Then in particular U S I U * + U N 1 U * = $2+N2. By the

theorem for the Jordan case proved above, UN1U*= N2 and US1 U * = $2.

Hence, for all h,k 6 H,

(PHNI h,k)H -- <Nl h,k>~ 1 --- <UNt h,Uk)~ 2

= <N2Uh,Uk>K 2 = (N2h,k)K2

= <PgN2h,k> H .

Therefore the strong nilpart A1 = P#N~IH is equal to the strong nilpart

A2 = PH N2I/4, and the theorem follows.

B~II and Helton 239

We now show that Theorem 1.1 can be refined to characterize the rela-

tionship between a Jordan operator T and its nilpotent part N.

T H E O R E M 1.6. Suppose T is coadjoint with strong nilpart A. Then T has

Jordan representation T = S + N with N = A if and only if A*A -- B2.

Proof. If T = S + N is Jordan, we have already verified that B2 = N*N.

Conversely, suppose T is coadjoint with strong nilpart A such that A*A = B 2.

Then W*B~/~ = A is the polar representation of A, and hence W* is a partial

isometry with initial space equal to (Range B~/~) - and final space equal to

(RangeA)-. Also, in Lemma 1.2, IIDfll~]-- IlWfll , and hence the map

F 6 H--* Df 6 K2 is a partial isometry with initial space equal to the initial

space of W, that is (RangeA)-. It then follows that the map ii: H---. K 1 is a

partial isometry with initial space equal to (Range A) . In particular it follows

that the map i: f 6 H ---, 6 K1 • K2 has range equal to all of K1 • K2. D

Thus, in the proof of Theorem 1.1 T is actually unitarily equivalent to

J = S + N and when H is identified with its image in K I ~ K 2 , we have

A = N .

§2. Spectral Factorization.

In this section we show that spectral factors of the symbol expansion

QT(s) for a coadjoint operator T are closely related to Jordan extensions of

T. Let QT(s) = Pl(s)*Pl(s) be a spectral factorization with P1 of the f o r m

Pl(S) = Vl+iSA1. Such exist by [R-R]. Then V 1V 1 = I

Ball and Helton 240

(a) (V 1A 1) VIA 1 = B 2

(b) V 1 A 1 (V1AI) = - i B I = T -

In other words N = VIA 1 satisfies conditions (1) and (2) in Definition 1.1 for

a nilpart of T. All of the work in this section is devoted to showing that N

frequently is a nilpart for T.

An ordinary Jordan operator T = S + N gives a nice example of the fac-

torization phenomenon. The nilpotent part N of T appears conspicuously in

the factorization

Qr (s) = (I + isN)* (I + isN).

A dramatic illustration of how the converse can be true comes by way of a new

proof of Theorem I of [H1]. It says that if T and T* are coadjoint then T

is Jordan and for the discussion here we make the added assumption that

Def(T) = (0,oo). To begin the proof let

P(s) = I + isN

be the outer factor of Q-r(S) and

R(s) = V2+isA 2

be the ,-outer factor of QT*(S)= QT(S) -l. Since p - i p , - i = R*R we have

PR*RP* = I which implies that the outer function P(s)R(s)* equals a (con-

stant) co-isometry, namely, V2. So

(V2 + isA2)*V2 R(s)*V2 P(s)- V2V2 = Z , ( - i sN) k. k=0

A first order polynomial equals a power series, which can occur only if N k = 0

for k >/ 2. Thus we have shown that N is indeed nilpotent. To finish the

theorem set S = T - N and use the argument [HI] (it's not trivial) to show

Ball and Helton 241

that S - S * and SN = NS.

The main theorem of this section is:

Theorem 2.1. Suppose T is coadjoint, D e f ( T ) = (0,n), and

Qr (s) -- (I + isA)* (I + isA)

is the outer spectral factorization of Qr(s). Then [CT(A+A*)]A = 0. In par-

ticular if Def(T) = (0,0), that is, if range A is dense, then

T*(A+ A*) = (A+ A*)T

and so A is a nilpart for T (with A*A actually equal to B2).

A purturbation argument allows us to use this theorem to produce a

strong nilpart for T and consequently by §1 a Jordan extension. This then

proves Theorem I.

Proof. For any spectral factorization of the above form, simply equating

coefficients of s and of s 2 gives

A - A * = T - T * ,

Since T is coadjoint, 0 = 1/2 C3(I)

A*A = B2 = 1/2 C2(I)

= CT(A*A). Use the general identities

CT(XY) = CT(X)Y + XCT(Y) - X CT(I)Y and CT(X*) = CT(X)* to obtain

0 = CT(A)*A + A*CT(A) - A*CT(I)A

= CT(A)*A + A*CT(A) - iA*(A-A*)A

= CT(A)*A + A*CT(A)- 2CT2(I)A + 2A*C2(I)

= CT(A*- 2 CT(I)) A + A*CT(A + 2CT(I) )

= 1/2 CA + 1A A*C,

Ball and Helton 2,42

where C = CT(A+A*). Hence

C(I + izA) = (I + i2A)*C

holds for all complex z. So whenever (I + izA) is invertible

(2.1) (I+ i2A)*-IC = C( I+ izA) -I.

The point of the proof is that the outerness of I + izA forces C(I-4-izA) -1

to be analytic and bounded in the upper half plane while equation (2.1) forces b

it to be bounded and analytic in the lower half plane. So Liouville's theorem

implies C(I+izA) - 1 = C which is the conclusion of Theorem 2.1. The main

work left in the proof is to insure the invertibility and boundedness of

(I + izA) -1 and this will prove nontrivial.

First we demonstrate I+ i zA is invertible for all z. Since O r ( z ) =

(I- izA*)(I+izA) =e-izT'eizT, we see that e-izTe izT* ( I - izA*) is a left

inverse of (I+izA) for all z. For z in a neighborhood of 0, I+ i zA and

I - i z A are each invertible by a Neumann series expansion, and hence on this

neighborhood

I = ( I + i z A ) ( I + i z A ) - l ( I - i z A * ) - l ( I - i z A *)

• - i z T i zT* = ( I+lzA)e e ( I - izA*).

But the right-hand side of this equation is entire, hence the identity continues

to hold for all z. Hence I+ i zA also has a right inverse, and so is invertible

for all complex z.

The proof now uses some lemmas, the statements and proofs of which

soon follow. By Lemma 2.3, there is a 8 > 0 such that

( I+isA)*(I+isA) >/ 8C2(I) for s real.

Ball and Helton 243

Since I + izA is outer, this inequality persists into the upper half plane [R-R];

that is,

1 IICT2(IY/2 (I+izA)-lH 2 ~ ~ for Imz > 0.

Now by Lemma 2.4,

IIc(I+izm)-lll .%< kllm(I+izm)-lll ~< kllC-~(I) '/-' (I+izA)-ll[

~ M = 8-~ ' for I m z > O .

Then also

|* II c (I - izA)-tl[ = II 0 + i~A)- C II = II c (I + izA)-lll

by equation (2.1), so it is ,%< M, for I m z > 0. We conclude that

C( I+ izA) -1 is a bounded entire function and so the theorem follows as previ-

ously described.

The next step is to give the lemmas needed in the proof of Theorem 2.1

and needed to do the subsequent purturbation argument. The A arising from

Theorem 2.1 satisfies A*A = B2 exactly, when in fact we need to satisfy, the

strong nilpart condition, where (1.4) is a strict inequality. The way we obtain

(1.4) is to perturb the inner product ( , ) on H to a new inner product in

which T is still coadjoint. We apply Theorem 2.1 in this new setting. The A

obtained there turns out to be a strong nilpart for T. When our general pertur-

bation argument is applied to the differential equations example it amounts to

perturbing q to q - 8 on the original space H. What follows is a string of

lemmas.

LEMMA 2.2. Let T be an operator on H. Let Y be any self-adjoint opera-

tor satisfying

Ball and Helton 244

T * Y - - Y T , a n d I ' Y > ~ e I , e > 0 .

Define a new inner product [.,.] on H by

[x,y] = ( (1-Y)x,y>

Let [ I be the Hilbert space

coadjoint as an element of L (/7/)

of L (H) .

Proof. Since I - Y > / ~ I , ~ > 0 , t h e H -

to equivalent norms. For X any operator on

adjoint

for x,y 6 H.

H w i t h the [.,.]

if and only if T

X[*]= ( i _ y ) - l x * ( i _ y )

= i (XT-TI*IX). Then

inner product. Then T is

is coadjoint as an element

and

H, let X [']

and set Ci-(X)

T - T I*] = ( I -Y) -1 [ ( I - Y ) T - T * ( I -Y) ]

= ( I -Y) -~ (T-T*)

(here we used the hypothesis T*Y = YT), and

C~(I) = T I * I ( T - T [*] - ( T - T [ * I ) T

-- ( I - Y ) - ~ [ T * ( T - T *) - (Y-Y*)T]

= (I--y)-ICT2(I).

Inductively, one can show

C~-(I) = ( I - y ) - I c - p ( I ) for n = 1,2 .....

/:/-inner products give rise

denote its L ( t ) ) -

Ball and Helton 245

and hence T is coadjoint as an element of L(H) (i.e., C3(I) = 0) if and only

if T is coadjoint as an element of L (/7/) (C3(I) = 0).

LEMMA 2.3. If T is coadjoint, then there is a 8 > 0 such that

QT(s) >/ 8 Cx2(I) for all real s.

Proof. Choose 8 > 0 sufficiently small so that I - 8 CT2(I) >/ d , for

some ~ > 0, and set Y = 8CT2(I). Then Y satisfies the hypotheses of

Lemma 2.2, and, in the notation there,

Qx(s) ___ e-isTt*JeisT = I + ( I - y ) - I C T ( I ) s + 1,6 ( I - -y) - IcT2(I )s 2

is positive-definite in the //-inner product for all real s:

[Q1,(s)x,x] = ((I-Y)Q1-(s)x,x) > /0

The above identity shows that this is equivalent to

Qr(s) = I + CT(I)s + ~/2 C-~(I)s 2 > /Y = 8 CT2(I)

for all real s. The lemma follows.

LEMMA 2.4. Suppose T is coadjoint with Def(T) = (0,m)

Qr(s) = ( I - isA*) (I + isA)

and

is the spectral factorization of QT(S). Then there is a constant k < oo such

that

CT(A+ A*) 2 ~ kA*A.

Proof. Set C = Ca-(A+ A*) as before. Then C - i Cr2(I)

= CT(A+ A * + T - T * ) = 2CT(A) since A - A * = T - T * . Also

iCr2(I) = 2iA*A, hence C = 2(CT(A) + iA*A), and

Ball and Helton 246

C 2 = C*C = 4(CT(A)* --i A ' A ) (CT(A) + i A ' A )

= 4 ( [ T * A * - A * T ] [ A T - T * A ] + [ T * A * - A * T ] A * A

+ A * A [ A T - T * A ] + A*AA*A)

since CT(A) = i ( A T - T * A ) by definition.

= 4 ( T ' A ' A T - T*A*T*A - A ' T A T + A*TT*A + T*A*A*A

- A*TA*A + A*AAT - A*AT*A + A*AA*A).

Now, by equation (1.6),

T ' A ' A T = ~,~ T*C2(I)T ~< I/2[ITI[2C2(I)

= IITtl 2 A*A

and hence, by the result of Douglas [Dou], there is an operator Z with liZll IITII s u c h that

AT = ZA, T ' A * = A'Z*.

The above becomes 4 A * [ Z * Z - Z 'T* - TZ + TT* + Z ' A * - TA* + AZ

- AT* + AA*]A ~< k A * A with k - 411['111 < ~'.

L E M M A 2.5. Let T be a coadjoint operator on H and let Y be a self-

adjoint solution of T*Y = YT with ~I ~< I - Y for some ~ > 0. Let [ / be as

in L e m m a 2.2, and I" the coadjoint operator T considered as acting on /7/i.

Then

Proof.

Def(T) = Def(T) ,

By possibly considering

Def,(~') = Def , (T) .

that Def(3") = Def(T) . For 0 ~ "0 ~ 1, let H ,

- T rather than T, we need only show

be the Hilbert space H with

Ball and Helton 247

the inner product

(x,y > ~ = ( ( I - 'oY)x ,y>

and let Tn be the operator T considered as acting on H,~. By Lemma 2.2,

T, is coadjoint, and hence Q r ( s ) has an outer spectral factorization

Qx(s ) = (V, + isA,~)V] (Vn + isA. 0)

([*] = Hn-adjoint). Express the identity in terms of the H-adjoint,

Q r ( s ) = (I-flY) -1 (Vn + isAn)*(I - flY)(Vn + isA,).

Also, by the proof of Lemma 2.2

QT, (s) = (I - flY) -1 [QT(S) - "0Y].

Hence

(2 .2 ) QT(S) - - flY = [(I-r lY) ' /~ '(V,+isA,)]*[(I- '0Y)v2(V,+isAn)].

Since V, + isA, is outer as an L(Hn)-valued function and (I-'0Y) '/2 is an

isometry of Hn onto H, it follows that (I-rlY)'/2(V,+isA0) is an outer

L(H)-valued function. Next we argue that in general, the map

Q(s) = A(s)*A(s)-- . A(s) which maps a nonnegative operator-valued func-

tions to its outer factor (which, if normalized to satisfy A(i) i> 0, is unique)

is continuous in the topology of supremum of the operator norms of the

coefficients. To see this, one reduces the situation to the case of bounded

operator-valued functions as in [R-R] by multiplying by a scalar outer function;

the continuity in this setting with the topology of operator supremum norm can

then be seen most conveniently from the proof of Sz.-Nagy-Foias ([N-

F],p.202). It follows that both Vn and A o are continuous functions of rl in

Ball and Helton 248

operator norm.

We first

(2.2),

and hence

show that d i m c o r a n V n = d i m c o r a n V for 0~<r j~< l . From

Vn( I - ' oY)V n = I - t r Y , for 0 ~< rt ~< 1,

kerV. = (0). Since the above also shows ( I -~Y) 'z 'V. ( I - t r Y ) -'/2 is

an isometry, it follows that V n is similar to an isometry and hence has closed

range. Therefore V, is semi-Fredholm for all 7. It is well-known [K] that the

semi-Fredholm index i(X) = dim k e r X - dim coran X is constant on con-

nected components of the semi-Fredholm operators, and hence

dim coran V I = - i ( V 1) = - i(Vo)

= dim coran V (since V0 = V).

The analysis for dim coran A n proceeds similarly. By (2.2),

A~ (I - nY) A n = '~5 CT 2 (I).

We recall the K2-metric on Hintroduced in Lemma 1.2:

< x,y >K2 = < 1/2 CT 2 (I) x,y> H'

From the line above, we see that (I-'0Y)~/2A~ extends to define an isometry I ,

of K2 onto r an [ ( I - ' oY) lnA, ] -. Hence I , , as an operator from K2 to H,

is semi-Fredholm; again by the continuity of the semi-Fredholm index,

dim c o r a n A 1 = dim coran It = dim coran I0 = dim coran A.

Hence Def(T) = Def(T), and the lemma follows.

Ball and Helton 249

We now have all the ingredients for the proof of the main theorem of this

paper.

Proof of Theorem I. Assume T is coadjoint and Def(T) = (0,0). (If

Def,(T) = (0,0), then Def(-T) = (0,0) and the following argument applies to

-T. ) Choose 8 > 0 sufficiently small so that I--8CT2(I) > el for some ~ > 0.

Then Lemmas 2.2 and 2.3 apply with Y = 8CT2(I). Hence, if T is the opera-

tor T considered as acting on /z/ (notation as in Lemma 2.2), then T is coad-

joint and Def('F) = Def(T) -- (0,0). By Theorem 2.1, if Qi-(s) has outer spec-

tral factorization (I + is,~) 1"1 (I + is.~), then

Tt*I(A + A [.1) = (,~ + ~[*])T.

Writing this in terms of the H-adjoint, we get

(I-Y) -1 T*[(I-Y)A + A*(I-Y)]

= ( I -Y)-~[ ( I -Y) , /~ + A * ( I - Y ) ] T ,

which simplifies to

T*(A+ A*) = (A+ A*)T

where

A = (I-Y),/~.

Hence A satisfies condition (1.3) of Definition 1.1.

Next we note that since ,~.-.~I*] = T_TV], we have

. ~ - (I-Y) -1 ,~*(I-Y) -- ( I -Y)-1(T-T *)

o r

Ball and Helton 250

A - A* -- T - T * ,

so A satisfies condition (1.2).

Since ~t*l~ --~h C~(I) = lh(I-Y)- lC~(I) , we have

= ( I -Y) -1 B2. Hence

A*A = ,~*(I-Y)2h~< ,~*(I-Y)A = B2,

( I -Y) -1 A * ( I - Y ) h

and condition (1.1) is satisfied. Finally, since A * ( I - y ) - I A = B2 by the

above, A* ( I -Y) -~/2 = B~/2X for a contraction operator X with

Ran X C [kerB~/q (see [Dou]), or A* = B~/~X(I-Y) '/2. Hence, if W is as in

Definition 1.1, W = X(I -Y) '/~. Hence

I - W*W = I - (I-Y)'/~X*X(I-Y) '/~

>~ I - ( I - Y ) = Y = 8B2

and condition (1.4) follows with k = 1/8. Hence A is a strong nilpart for T,

and by Theorem 1.1, T has a Jordan extention. Theorem I now follows.

Now we present some partial results on a couple of approaches toward

removing the Def(T) = (0,0)-hypothesis from our main theorem.

Reduct ion of general case to (O,n) or (m,O) case. In the concrete

examples to be presented in §3, there is always a subspace H0 of H which is

invariant for T (corresponding to zero boundary conditions at the end points

of the interval), and a coadjoint extension T1 of T to a slightly larger space

H~ (corresponding to free boundary conditions at the end points). The exam-

ples also suggest that Def(TlH0) = (n,0) while Def(Tl) = (0,m). The idea is

to try to find abstract definitions of H0, HI and TI; then if one can prove Tl

is coadjoint, the coadjointness of T follows automatically; since H0 is a

Ball and Helton 251

"large" subspace of H invariant for T, proving that TIH o is coadjoint makes

the coadjointess of T very plausible.

Our candidate for /40 is given by the following.

PROPOSITION 2.6. The subspace

Ho = {g E H: there is a Kg > 0 with

[(Blf, g)l 2 ~< Kg B2f, f

for all f in H}

of H is invariant for T.

Proof. Define ( ' , ')L by <f,g)L = (B2f, g). Use the definition of B2 to

obtain

(I/2(T-T*)f,Tg) = ([B2 + 1d (T-T*)T]f,g) /

= ( f,g)L + ( 1/2 (T-T*)Tf, g )

Hence if g 6 H0and f 6 H

i Bl f,Tg)l ~< IlfllL IlgllL + Kg<B2Tf, T f ) '/2 I<- 7

and since T*B2T ~< [[TII2B2 we get

i I<- Blf, Tg>l {llgllL + KgllTII}llfllL

Thus Tg is in Ho.

We now verify that Ho in the context of the differential equations exam-

ples to be given in §3 (excepting the case of Periodic boundary conditions) is

just

Ball and Helton 252

Soo = {f 6 H : f(0) = f(1) = 0} ,

as desired. The computation

<1/2 (T-T*)f ,g )H = V2 f (f~'- f~)dx

= ff 'dx- v2 f(1)g(1) + v2 f(O)g(O)

shows that, for a given g 6 /-/, the linear functional f---. ( lk(T-T*)f,g)H

extends to define a bounded linear functional on L 2 if and only if both g(0)

and g(1) vanish. Thus H0 = So0.

We suspect that one may "dualize" about H0 to obtain H1 but we have

not carried this out successfully.

Remarks on Def(T) = (0,n) eoadjoint operators. The main open question

,for such a T can be phrased: Does

(2.3) (T2f, f) = (f, T2f>

for any f in kerA* (where

that Theorem 2.1 says that if

(T*(A

A is defined as in Theorem 2.1)? The reason is

+ A*)f,f> = ((A + A*)Tf, f)

for f in kerA*, then A is a nilpart for T, and hence (via our perturbation

argument), T is sub-Jordan by Theorem 1.1. The left side equals

(T* (A-A*)f,f> = (T* (T-T*)f>

while the right side equals

( ( -A+A*)Tf,f> = ( ( -T+T*)Tf , f )

Ball and Helton 253

Another open question is: Does range A contain the space H0 defined in

Proposition 2? In our examples (§3) this is always the case; furthermore the

equality (2.3) holds for all scalar multiples of real valued f in H ~ H0. An

abstract version of this might be easier to work with than something which

requires determination of kerA*.

We conclude with a test t o determine if n = 0 for a T with

Def(T) = (0,n). The test is: n = 0 i f and only i f there is no x so that

(QT-(S)X,X) = (x,x)

for all real s. To prove it simply note x E ker A* if and only if

(QT,(S)X,X) = ( ( I - i s A * ) - I x , ( I - i s A * ) - l x ) = (x,x) .

P A R T II. E X A M P L E S

§3. Differential Operators

We have already seen that one coadjoint operator is T = M× on a Sobolev

space whose inner product comes from a Sturm-Liouville operator. We also

used Sturm-Liouville disconjugacy theory to show that this operator M× has a

Jordan extension. Then we proved, in the main theorem of the paper, that

very general classes of coadjoint operators have Jordan extensions. In part (a)

of this section we demonstrate that this theorem implies a disconjugacy

theorem as stated in the introduction for very general Sturm-Liouville opera-

tors. The book by Reid [R] is the most comprehensive reference for classical

factorization theorems (see Theorem IV. 7.4). Our algebraic approach is intrin-

sically more general as well as being cleaner. Part (b) of the section gives a

represenation for a sub-Jordan operator as a multiplication operator on a

Ball and Helton 254

Sobolev space. In part (c) we give examples of coadjoint operators obtained as

Mx on Sobolev spaces with different boundary conditions. This gives consider-

able intuition and strong evidence for our conjecture in the introduction on the

relationship between defect and boundary conditions.

(a) Factorization of Sturm-Liouville operators. We shall be interested in

factoring second order differential operators as the product l)*I) of first order

operators 13. Our main theorem roughly says that if the Hermitean form given

by L on a space S of functions is positive and if M× on the space has defect

= (0,0), then L can be factored as D*PD. This subsection begins with a

lemma which describes abstractly what we mean by a first order operator D.

The lemma is closely related to representation theorems of Jorgensen and

Muhly [J-M] and Dorfmeister and Dorfmeister [D-D] who start with somewhat

different hypotheses.

LEMMA 3.1. Suppose S~ is a bounded self-adjoint operator represented as

multiplication by x on a Hilbert space Ki = ~, L2(mij) w h e r e {mij}j=l , 2 .... is a J

collection of measures such that mi,i+l << mi,i (<< = "is absolutely continu-

ously with respect to") for i = 1,2. (We allow the measures mi,i to be zero for

j sufficiently large; we regard elements of K i as column vector-valued func-

tions.) Suppose also that I3 is a closed operator from Ki into K2 with

domain D dense in K1 such that

S I : D - - - ' D and 13Sif= (S213+F)f ( f6 D)

for some operator F 6 L(KbK2) satisfying FSI = S2F. Let N be the smal-

lest cardinality (finite or ~o) of a set of vectors S = { ~ : l , . . . ,~N}

(S = {~:1,~:2,..-} if N = ~ ) C D such that

(*) V{p(S1)~:i: p a polynomial, ~i 6 S}-

Ball and Helton 255

is a core for 13; let s c denote the matrix-valued function with N columns

= [~1 ,~ :2 , ' ' " ,~N]

(s c =[~,~%,...] if N = co).

Let Co 1 = {f 6 KI: each coordinate of f is a continuously differentiable func-

tion on sp(S~), all but finitely many coordinates of f vanish on sp(S1)}.

Then

(1)

(2)

Do = {~:'f: f 6 Co 1} is a core for D and

there exist matrix-valued functions 0 = 0(x)

such that, for g --- {:f 6 Do

(13g) ( x ) = ( 1 3 , f ) ( x ) = 0 ( x ) f ( x ) + F ( x ) , (x) [ d--~-f ]

that is, formally 13 is the differential operator

13 = 0~ :-~ + F~: ~ ~:-1.

Proof. By iterating the identity

F S~ = $2 F, we see that

13S{ ~ = S~'D + nS~-IF

and hence, for any polynomial p,

(3.1) 13p(S~) = p(S2)D + p'(S2)F.

and F -- F(x)

(x) ,

D S ~ = S 2 D + F and using that

(n - -1 ,2 ,3 .... ) ,

Since I3 is closed, this identity is also valid if p is continuously differentiable

on sp(Sl). (Since FS1 = S2F, RanF c E(o-(S1))K2, where E is the spectral

measure for $2 by well-known results for self-adjoint operators [Dix].) Hence,

Ball and Helton 256

KflE if f = 6 C 1, t h e n ~ : . f = fl(x)sCl(x) + . . . + fn (x )~ , (x ) 6 D. Since

(.) is a core for D, it follows that Do = {~: ' f : f 6 C 1} is a core for 13. Since

F intertwines S1 and $2, by general facts about self-adjoint operators ([Dix]),

F is multiplication by a matrix-valued function F(x). For i - 1,2 .... , and

j = 1,2 ..... N, define the scalar-valued function 0i,i by

0ij(x) = [I)~:j]i = the i th coordinate of 13~:j,

and define 0 to be the matrix-valued function with

[0(x)]~j = %(x ) .

Then for f 6 Co 1,

13(~:f)(x) -- 13(ft~:l + ' " + fn~:n)(x)

= ( f l ( x ) (13{?O(x) + " " + f . ( x ) (I3~:~)(x))

+ (f{(x)F(x)sCl(x) + • • • + fn(x)F(x) sen(x))

(using (3.1))

= 0(x) f(x) + F(x) , (x) I d-~- fl (x) •

The lemma follows.

Remark. In the context of differential geometry, I3 is called a "connec-

tion" and O is called a "connection matrix" for 13 with respect to the frame

for K1 and the frame induced by the identity matrix for K2 (see [W]).

Ball and Helton 257

Now we give our factorization theorem. Suppose that ( , >, is an inner

product which makes a space H o f functions on [0,1] a Hilbert space and on

which T = Mx is coadjoint. For example any positive definite Sturm-Liouville

operator L0 induces such a sesquilinear form on a space of functions with

appropriate boundary conditions. If De f (T)= (0,0), then by §2 the outer

spectral factorization of Qr yields a nilpart A for T. The next theorem

describes how the nilpart gives directly a factorization of ( , >, in other words

it gives a factorization for L in weak form To keep things simple assume

C~(I) has no null space.

THEOREM 3.2. Let A be a nilpart for the coadjoint operator T o n H , and

assume KerBS= (0). Let K 1 and K2, $1 and S2, il and i2, and

D: H- - . K2 be as in Lemma 1.3. Let U: H- - . K1 (9 K2 be defined by

ii1 1 U: f---. (note that since KerB2 = (0), U is one-to-one) let K ' - - {Uf: [izf]

f 6 /4}- (closure in K1 (9 K2), and S' - S1 (9 S2I K'. Define the operator

I5: KI ---" K2 with domain D = {Uf: f 6 H} by I ) U f = Df. Then

(1) 15 is a closed operator,

(2) D is invariant under S' and I ) S ' f = (S2D'+ F)f for all

(3)

(4)

f 6 D where F:

F S ' = S2F.

UT = S'U,

:1 - , is in

There is a positive operator Q on

such that, for all f 6 /4,

L (K',K) and satisfies

K' commuting with S'

][fl[~ = <QUf, U f>K'-t-1115 U fll~2

Ball and Helton 258

(5) If Def(T) = (0,0), and A arises from the outer spectral fac-

torization of Qr as in Theorem 2.0, then K1 degenerates to

the {0}-space. Consequently K ' = K2 and Q = 0 so

(f,g>/4 = (13 U f,I~ U g)K2.

Example: In the Sturm-Liouville case with (0,0)-defect, (g,h)H = 1

f0 (Lg,h),Tdx, B 2 = L-1P and U is inclusion, so (Ug,Uh)K2 = <B2g,h)H =

1

f 0 ( P g,h)n dx so our factorization result becomes

1 1

Y (Lf, g)ndx = <U-lf, U- lg )n = y (PI)f, Igg)ndx 0 0

as desired.

Proof. The proof is completely analogous to that which will be given for

Theorem 3.3. We leave the details to the reader.

(b) Representations for sub-Jordan operators. This subsection demon-

strates that any sub-Jordan operator corresponds to a Sturm-Liouville type

model. The next theorem (along with the spectral theorem for self-adjoint

operators) sets down the construction.

THEOREM 3.3. Suppose T is a sub-Jordan operator on H. Then there is a

Hilbert space K = KI • K2, a self-adjoint operator S on K with operator

o F matrix representation S = , a nilpotent operator N on K with operator $2

matrix representation N = [~ :], a closed operator I5 : K,--- K2 and a projec-

tion operator P on K2 commuting with $2, such that

Ball and Helton 259

(1) the domain D of I) is dense in K~ and is invariant under

SI,

(2) 0S1 f = (S21)+PF)f ( f6 D) ,and

(3) there is an isometric transformation U: H---, K such that

{If] / U H = • f 6 D,k 6 (I-P)K2 D f + k

and

0] ul"u*luH= lug.

$2

Remark. If we identify a typical element

k 6 ( I - P ) K 2 ) w i t h [SI 6Kl~K2, then H is

Dom(D*D) ~ (I-P)K2 in the norm

IIIS]II~= ( I+D*D)f , f Kl+llkll~2. tK]

[fl U* (f 6 D, I)f+k

the completion of

As the action of [) is given by a first order differential expression by Lemma

3.1 (when K~ and K2 are taken to be the spectral representations for $1 and

$2 respectively), the operator L -- I + I3"I3 can be thought of as a Sturm-

Liouville operator. In this representation, the action of T becomes

0]i:l (I-P)F $2

Ball and Helton 260

In this way we see that the most general sub-Jordan operator is a combination

of multiplication by x on a Sobolev space and of a Jordan operator.

Proof of Theorem 3.3. For conciseness, we use the machinery already

developed in §1. (It is also possible to proceed directly.) Suppose T is sub-

Jordan and A = PHNIH is the strong nilpart coming from a Jordan extension f ~

of T. Let KI, K2, SI, S2, N = ] ~ :1 and D H - ' K I be as given by L -

Lemma 1.3 and the proof of Theorem 1.1. Define an operator I): Kl--" K2

with domain D equal to il H (notation of Lemma 1.3) by the formula

13(il f) = g E K2, where ]]gl]h2.2 = min{[]Dkn2. 2 ' i l f = ilk} ,

Then it is clear from the definition and elementary Hilbert space theory that I7)

is well-defined.

We show that 13 is closed. Suppose i,fn K~h and D(i,fn) /<2 z as

n---,oo. Choose t',~ so that I l l 'hi ]h=-[[ i l I 'n[Ik, +[]Dfnl[~" 2 =min{] lgl l~t"

i lg = ill'n}. Then in fact D( i l fn) = Dt'n, and i l fn = ili'n. Hence it is no loss

to assume that fn = fn in the first place. Then []fn -- fkl]/2t =[]ilfn - ilf~k~

+ llDfn - Dfkll~2 is Cauchy, and hence f, H f' Since il and D are continu-

ous, it follows that i t f = h and D f = £. Since l)it fn = Dfn for all n, it also

follows that I)(il f) --- D f = 2. Hence 1) is closed.

Let Q be the orthogonal projection of K 2 onto the subspace {k ¢ K2:

k = D f for some f ¢ H such that i l f = 0}. Since for any such k,

S2k = S2Df = (DT - i2)f, i2f= F(i lf) = 0, and i l T f = S l i l f = 0, we see

that RanQ is invariant under $2 and hence QS 2 = S2Q. We leave it for the

reader to check that all assertions of the theorem now follow if we set P = I-Q.

(c) The defect in examples. We next discuss the differential equations

Ball and Helton 261

examples in the scaler case to illustrate the possibilities for Def(T). In each of

the following examples we shall exhibit a class of factorizations

(3.2a) Qr(s) = ( v + isA)* (V+ isA)

of the symbol expansion of T such that V*A is a nilpart for T, namely

(3.2b) T*(V*A+A*V) = (V*A+A*V)T.

The factorizations are built from real-valued continuous solutions of the Riccati

equation

(3.3) fl, = q _/~2.

One characterization of outer (respectively ,-outer) for an operator-valued

polynomial P(s) -- V+ isA is that the quantity

(3.4) P(i)* P(i) -- ( V - A)* ( V - A)

is a maximum (respectively minimum) among all polynomials with P(s)* P(s)

equal to a given operator polynomial Q(s) for s real. Each well behaved solu-

tion /3 of the Riccati equation gives a factorization of Qr(s), but of particular

interest is the ~ which makes (3.4) the largest (respectively smallest) (within

this restricted class of factorizations). We are unable to prove that the

corresponding factorization is outer (respectively ,-outer) (i.e., extremal for

the class of all factorizations), but we strongly suspect it is. In each example

we compute the defects corresponding to factorizations with a general 8. In

some examples, the defects coming from the extremal /Ts are different from

those coming from a generic ft. These examples form the basis for the conjec-

ture in the introduction on. defects.

In certain cases, it is convenient to introduce an intermediate space H~,

so that (V+isA): H---* H1. In the first three examples T = Mx on a

Ball and Helton 262

Sobolev-type function space H. The inner product on each H comes from a

d 2 Sturm-Liouville operator L = dx 2 + q on some domain which makes L

self-adjoint. As these results will not be needed later we only state the results

and omit the details.

1

Example 1. H = S 0 0 with Ilfll~ = f J f ' P +qJf~ 2dx. Then L on

D o m ( L ) = { f 6 L 2 : f , f ' 6 AC, f ' 6 L 2and f ( 0 ) = f ( 1 ) = 0 } gives a n o r m on

Dom(L) whose completion gives H.

If /3 is any continuous solution of (3.3) then integration by parts shows

Ilfl[b,-- f0 I -/3 ffdx.

Let H l = { f 6 L 2' f 6 AC, f ' 6 L 2, f(0) --- 0} with

Define V: H---. Hi and A: H---. Hi by

V: f---. f

A:f---*g where [ d__d___/3]g=f and g ( 0 ) = 0 . t~lx i

Then one can show that (V,A) satisfies (3.2), and moreover

(dim coran V, dim coran A) = (1,0) .

A further computation shows that

II (v-m)fll~l = [If lib + Ilfl~2 + 2 f / 3 f? dx.

Thus a maximal (resp. minimal) /3 corresponds to that /3 which is maximal

(resp. minimal) among all continuous solutions of (3.3) in the pointwise partial

ordering on real-valued continuous functions. The maximal such /3 will have

either /3(0) = dO or /3(1)= dO and the minimal one will have either

Ball and Helton 263

/3(0) = - oo or /3(1) = -oo . In either case,

(dim coran V, dim coran A) =(1,0) .

1

Example 2. H = So, with ]lfll~ = al[f(1)~ + f0 {If'12 + qlf~}dx Then

D o m ( L ) = { f 6 L 2 f , f ' 6 AC, f " 6 L 2and f ' ( 1 ) + a l f ( 1 ) = 0 } .

Again if a continuous /3 satisfies (3.3) and also al +/3(1) >/ 0, then

Ilfl[b = [a~+/3(1)llf(1)~ + fo ] - /3 f~dx

If we choose /3

A : H - - - . H b y U f = f and A f = g where [d--d- /3]g=

d %

(3.2Yholds and (dim coran V, dim coran A) --- (0,0).

such that - a l =/3(1), then we can define V: H - - . H and

f and g ( 0 ) = 0. Then

For /3(1) > - a l , let H1 = {f ~ L2: f 6 AC, f' 6 L 2} with Ilf]l~ = fo

][d--d-x--/31f 2de + [a l+ /3 (1 ) ] l f (1 ) f 2. (Note that / 3 ( 1 ) + a l > 0 is needed to

1 %

show that this inner product is positive definite.) Define V: H - . HI and

A:H--*H1 b y V : f - - f and A:f---.g w ere = f and

d '%

g(1) = 0. % p

Then (3.2) holds and

(dim coranV, dim c o r a n A ) = (1,1).

that

For both cases described above (B(1) = - al,/3(1) > - a0, it turns out

1

It (V-A)fI[~, = Ilfl[~ + I]fl~2 - If(1) ~ + 2 fo/~ f? dx.

Ball and Helton 264

Thus a maximal (resp. minimal) ,8 again is one which is pointwise maximal

(resp. minimal). The minimal /~ will be the one for which ,8(1) = - al which

has defect (0,0). All other ,8's give rise to a defect of (1,1).

1

Example 3. H = {f E L2: f E AC, f 'E L 2} with Ilfll~ = fo {If'•

+ qlf~} dx - ao[f(O)~ + allf(1)~. Then Dora(L) = {f E L2: f,f' E AC,

f"E L 2, f ' ( 0 )+ao f (0 ) = 0 , f ' ( 1 ) + a ~ f ( 1 ) = 0 } where ao and al are real

numbers. If ,8 is a continuous solution of (3.3) satisfying ,8(0) + ao ~< 0 and

,8(1) + a I /> 0, then alternatively (via an integration by parts)

IIfl~ = 3"o[ dl-d-~- x - ,81 f[2 dx - Us(o) + ao] f(o)[z + [,8(1) + a,]'f(1)[ 2

If / 3 ( 0 ) + a 0 = 0 , define V : H ~ H and A : H ~ H by V : f - - f and

an0 A : f ~ g where % i

(dim coran V, dim coran A) = (0,1). Similarly if /3(1) + al = 0, define

and g(0) = O. Again (3.2) holds and (dim coran V, dim coran A) -- (0,1).

However, if both /3(0) + a0 < 0 and /3(1) + al > 0, define a space

H l = { f + c 8 1 : f E AC, f 'E L 2 , c E C} where81(x )=

[°l IIfllb, = f l T x - ~ f[ 2dx - ~ ( 0 ) + ao]lf(O)[ 2

0, x ;~ 1, with 1, x = l

+ [/3(0) + a~]lf(1)[ 2.

Ball and Helton 265

Define V: H ---. Hi and A: H ---* H~ by V: f ---* f and A: f ---* g + ~ l where

[ d ~ - / 3 l g = f ' g ( 0 ) = 0 and g ( 1 ) + c = 0 . Then (3.2) holds and P

(dim coran V, dim coran A) = (1,2). In all of these cases (i.e., no special res-

trictions on/3(0) or /3(1)), it turns out that

II (v-m)fll~ = Ilfll~ -~lfll~2 + If(0)12 - I f ( l ) t 2

+2 f / 3 f f dx .

Hence again a maximal /3 corresponds to one which is pointwise maximal. It

seems reasonable to assume that this occurs for /3(0) - , %, which gives rise

to a defect of (0,1). Similarly a minimal /3 should be the case/3(1) = - a b

which also gives defect (0,1). In all other cases the defect is (1,2).

Example 4. H = {f E L2: f E AC, f' E L 2, f(0) = f(1)} with IIf[l~ = 1

f0 {If'f + qlff}dx. Then Dom(L) = {f E L2: f,f' E AC, f" E L 2, f(0) --- f(1),

If /3 is a continuous solution of (3.3) satisfying /3(1) >//3(0), f'(O) = f'(1)}.

then

Ilfll~ = g l - / 3 f~ + [/3(1) - /3 (o ) ] ] f (1 )~ .

In the example, we cannot take T to be multiplication by x, since this opera-

tor does not leave H invariant. Instead we take T to be multiplication by a

real-valued absolutely continuous function 0 such that 0(a) = 0(b) and

0' E L 2. (Examples 1, 2, 3 could have been worked out for an analogous such

T; the results would be the same "mutatis mutandis".) Then H is invariant 1

under T, 1/2 C~(I)f -- L-l (0'2 f) (f E H) , and T iscoadjoint. If fn/3(s)ds

;e 0, then the boundary value problem for g, given h in L 2,

Ball and Helton 266

[d-~--/3]g=h' g(O)--g(1)

always has a unique solution. Hence we can define V: H---, H and -al 1 %

A : H - - " H by V f = f and A f = g where [d--d--/3lg =O'f and ~t p

g(0) = g(1). Then (3.3) holds and (dim coran V, dim coran A) -- (0,0). We

remark that if/3(0) --/3(1), so that Ilfll 2 = fo I - / 3 fl2dt, ther~ are no

[ d = 0, since by assumption L is nontrivial periodic solutions 4' of -~x

strictly positive definite, and hence we are in the above case where 1

f B(s)ds # 0. 0

1 If Jo-/3(s)ds ='0, then we are in the case where/3(1) >/3(0) and we

can define a Hilbert space H1 = {f 6 L2: f 6 AC, f' 6 L 2} with Ilf[121 =

1 [d-~ I f0 - / 3 f~ + [/3(1) -/3(0)]If(I)[ 2. Define V: g - . H, and A: H---, Hi

and A f = g where I d - d - / 3 l g =f,g(1)----0. Then (3.2) holdsand

J ~

b y V f = f

(dim coran V, dim coran A) = (1,1).

In all the above instances for the periodic case,

II (v-n)f[l~,l = Ilfll~ + Ilfl~2 + 2 f / 3 f f dx.

Hence a maximal (respectively minimal) /3 again corresponds to one which is

pointwise maximal (respectively minimal). The constraint on /3 is equivalent

to the condition that the expression

Ball and Helton 267

(3.5)

be independent of /3 for each f in H, together with /3(1) >//3(0). Intui-

tively, as /3 increases in magnitude the integral term in (3.5) gets large, so

/3(1) - /3(0) must decrease in order to keep (3.5) invariant. It follows (at least

heuristically) that for both extremal/3's, /3(1) =/3(0), which yields a defect of

(0,0).

If we view this example as based on the circle by identifying the point 0

to 1, then /3(0) ~/3(1) means that /3 has a jump discontinuity at 0 = 1.

From this point of view there is nothing special about 0 = 1; we could have

introduced a jump discontinuity of /3 (with positive jump) at any other point in

(0,1) and built a corresponding (V,A). Similarly jump discontinuities of /3

could have been introduced in the previous (non-periodic) examples.

It would be interesting to compute Def(T) for the above class of exam-

ples (including the vectorial case) in terms of the boundary conditions on the

space H. A general vectorial example (encompassing Examples 1, 2, 3) is to

take T = Mx on the Hilbert space of N-valued functions on [0,1].

H = { f E L~: fE ACN, f 'E L 2,0~*f(0) =0 ,0bf (1) =0} with

llfl~ = (Qa f(0), f(0))u + (Qb f(1), f(1))u

1

+ f {(Pf',f')u + (Qf, f)s} dx J

"0

Here Qa and Qb are self-adjoint operators on N, P and Q are self-adjoint

operator-valued functions with P positive definite, and Oa and Ob are opera-

tors on N. If in the above examples it is the case that a maximal (respectively

minimal) /3 induces the outer (respectively ,-outer) factorization of QT(s),

then a conjecture (alluded to in the introduction) consistent with the examples

Ball and Helton 268

is that either Def(T) = (0,0) or Def , (T) = (0,0) if and only if the boundary

condition for H at 0 is adjoint to that at 1 in the sense that rank[0~,0b]

= dim N ~ n, and r b = rank0b = n -- r,, where r~, = rank0~.

§4. Compressed Jordan Operators.

As described in the introduction an ultimate goal of our program would

be to describe those operators T which arise from a Jordan operator J on

K D H b y

e = P t t J Pt/ where = 0 , + 1,-4-2 . . . . .

Such a T will be called a compressed Jordan operator. We can' t solve the gen-

eral problem but do obtain the following rather modest proposition, which is

the main result of this section.

T H E O R E M 4.1. An operator T is the compression of some finite dimen-

sional Jordan operator if and only if T is Jordan.

Proof. By hypothesis, T = Ptt JItt where J is sub-Jordan (and hence

coadjoint) on K D H and K G H is an an invariant subspace for J. The

main step in the proof is an explicit formula for the symbol expansion QT of T

in terms of that of J. To obtain this, write

Set A(~) = r ~ + ~ r a n d l a b e l A o A o . oA as A n. Then

j _[o

Ball and Helton 269

and

Qj(s) = I e -isr*e isr e -is< e isA(O~) ]

e-iSA*(a) eis~ e-iSA*(c~) eisA(o0 + e-isT*e isT

[PII(s) P,2(s)]

[P2, (s) P22(s)] '

from which we get

QT = P22 - eisA*(°~) eisA(~x) ,

SO

(4.1) QT(S) = P22(S) -- P21(s) Pll(s) -1P12(s).

Since Qj is an operator pOlynomial of order 2 the Pij are also, so (4.1) says

that QT is a formal operator coefficient rational function of order (2,2). In

infinite dimensions it is hard to make use of this. However, in the finite

dimensional case, Pll(s) - l = (detPll(s)) -1 adjPll(s), and hence all entries of

the matrix function Qr(s) are rational. Since Qv(s) is also entire, it follows

that Qx(s) is a matrix polynomial.

The next step is to note that since T has the form PHJ ' lHwhere H is a

semi-invariant subspace for the Jordan operator J', the formula

f(T) = PH f(J ' ) ]H

defines a Cl(R)-functional calculus for T. Hence the Jordan canonical form

for T assumes the form

M - I T M = o- + j

Ball and Helton 270

where o- is self-adjoint and j2 = 0. The theorem now follows from the follow-

ing general proposition.

LEMMA 4.2. If the symbol expansion QT for the operator T is an operator

polynomial and T is similar to a Jordan operator, then T is Jordan.

Proof. Suppose M - 1 T M = o- + j where o- is self-adjoint, j2 = 0 and

o-j = jcr. Then eiSTM = M e is~ e isj, and hence

e isj* M* Qr(s) Me -iSj = e -is'T M* Me iS'T .

By assumption the left-hand side is a polynomial. Since the right-hand side

remains bounded as s tends to infinity through real values, this polynomial

must be a constant, or M*M commutes with or.

Write M in polar form U(M*M) ~/? with U unitary. Then

T = M ( c r + j ) M - 1 = U(cr+j ' )U* where j ' = (M*M)'/2j(M*M) -'/?. Thus T is

unitarily equivalent to a Jordan operator.

P A R T III. S E V E R A L C O A D J O I N T O P E R A T O R S

§5. Commuting Jordan and coadjoint operators.

In this section we indicate generalizations of the structure in the preceding

sections to higher dimensions. As we shall see there are two basic directions

for generalization; one involves "real" structure and we give an example which

shows that extensions of this type are closely related to conjugate surface theory

for partial differential equations. This is described in §5. The other basic direc-

tion involves "complex" structure and these operators fall into the framework of

Cowan and Douglas [C-D]; this is in §6.

Ball and Helton 271

Henceforth let J = {J~}~= I be Jordan operators (of order 2) on a Hilbert

space K. We call J a commuting collection of Jordan operators (C.C.J) if

(5.1) Jm J~, = J~, Jm

for all ~,,m. The first surprise is that a C.C.J. is more commutative than one

would think.

PROPOSITION 5.1. A commuting collection of Jordan operators in addition

to satisfying (5.1) satisfies

(5.2) Sg, Snl=SmS~, S~Nm=NmS~, N~Nm=NmN~

for all g,m.

Besides commutativity there is another natural restriction one can place

on a collection J of Jordan operators, namely

(5.3) N~?, N m = 0 for all V,,m .

A C.C.J. which satisfies (5.3) will be called a Jordan family (J.F.). That (5,2)

and (5.3) are not equivalent can be seen by considering two carefully chosen

nilpotent operators on C*, namely

fi00ip00 (5.4) N t = N2=

0 0

0 1

li ° ° il o00010 In this section we study C.C.J.'s and J.F.'s and the structure of their restric-

tions to invariant subspaces.

Proof of Proposition 5.1. It suffices to consider two commuting Jordan

o p e r a t o r s J l = S l + N 1 a n d J 2 = S 2 + N 2 . Set

Ball and Helton 272

Q j ( s I , s 2 ) = e-iS2J2 e-is,J1 eiSl J, eiS2J2

and Q j , ( s l )= e -is'J'

i(XJ~, - J~X) (z = 1,2). Since Jl

Cj1 commutes with Cj 2. Hence

Qj (s,,s2) = e s2q2 (e s'cJ' (I))

= e s2cJ2+s'cJ' (I)

1

islJ 1 e , and define Cj~: L ( K ) - - - * L ( K ) by C j ~ ( X ) =

and J2 commute, it is easily checked that

o cjn2(1)s~s~ .

Since each J~ is Jordan, Cf~ (I) = 0 for k >/ 3, and we see that Qj is a poly-

nomial in sl and s2. From the identity

* i s2S2~ ' (1 + is2N2)Qj (Sl,S 2) (1 - i s2N 2) = e Qj I (s 1) e 's2s2 ,

- i s2S 2 ~ z ~ eiS2S2 we see that e I.~Jl~S 1) is a polynomial in s2 which, for fixed values of

sl, is bounded as s2 ~ -+ ¢o through real values (since $2 is self-adjoint).

Therefore this polynomial must be constant, and so $2 commutes with Qj,(sl)

for all %. It follows that $2 commutes with the coefficients of Qj,, namely

i ( N I - N ~ ) and N~N1. Then by Lemma 1.5, $2 commutes with Ni. By sym-

metry $1 commutes with N2.

Since the S's and the N's commute, Qj may be written as

Qj = (l-islNl*) (I-is2N2) (I+is2N2) (I+islN1)

= (I-is2N2) (l--islN~) (I+islNl) (I+is2N2) .

Ball and Helton 273

Equate coefficients of s~s2 to obtain

- N[N2 +N1N2 + N2N1 - N2NI

= - N]N[ + N2N1 + N1N2 - NIN:

o r

(a) N[N; + N2N, = N;N[ + N,N2.

Equate coefficients of s~s~ to obtain

(b) N[N2N2N 1= N]N[N,N2.

Again by Lemma 1.5, it will follow from (a) and (b) that NIN2 = N2Ni if we

can show that N~N 2 and N2N ~ are nilpotent. By symmetry we need only

show N1N2 is nilpotent. Multiplying (a) by N2N1 on the right and using

N2 2 = 0, we see that

which by (b) gives

N[N]N2NI + (N2N,) 2 = N]N[N2N1,

(N2N1) 2 = N2N[(N2N 1 - N1N2) .

Since N f = 0, we then get

(NzNI)*2(NxN1) 2 = 0 ,

and so (N1N2) 2 = 0 as desired. As indicated above, N1N2 = N2N1. Since

S1S 2 - - 82S 1 = [J1J2 - - J2J1] - - [ N I S 2 4- S 1 N 2 - - S 2 N 1 - - N 2 S I]

- [ N I N 2 - N2N1] ,

Ball and Helton 274

Sl commutes with $2 as well. Proposition 5.1 is proved.

We now discuss sub-Jordan operators. Begin by setting conventions. A

boldface capital letter, e.g. T, will always indicate a collection of k operators

{T~} k on a Hilbert space. We say that the family T on H is extendedby J on

K provided H is a subspace of K which is invariant under each JR in J and

T~, = J~[ft for each ~,.

It is trivial to prove.

P R O P O S I T I O N 5.2. If T can be extended by a commuting collection of Jor-

dan operators J, then T is a commuting collection of coadjoint operators

(C.C.CA.).

This suggests

Conjecture 5.1. If T is C.C.CA., then T has a C.C.J. extension J.

Next consider a T which extends to a Jordan family J; the additional

restriction (5.3) on J adds a restriction on T which is most intuitively seen

in terms of its symbol expansion. If T is any commuting family define its

symbol expansion QT(S) by

Q T ( S ) e - i S k T ; i s l T ; e iS l T ' e iSkTk " " " e - - - ,

When T is a C.C.CA.

Jt QT(S) = ~ , ~j, . . . . jkSI

J l ' Jn maxl j 1 . . . . . jk}~<2

• . . SJk k

If T can be extended to a J.F., then it is easy to see

)l Jk (5.5) QT(S)= 2 /~J l , . . .Jk S] " ' " Sk "

J] . . . . Jk j l + j 2 + . + j k ~ < 2

Ball and Helton 275

Equivalently,

(5.5') CTj. ° C j m o Cx~ (I) = 0

(j,m, Z = 1 ..... k) .

Call a T satisfying (5.5) a coadjoint family (CA.F.). This plus the example

(Theorem 5.4) which follows leads us to

C O N J E C T U R E 5.2. A family T of operators has a J.F. extension J (i.e., J

satisfies (5.3)) if and only if T is a coadjoint family (i.e., T satisfies (5.5)).

These conjectures 5.1 and 5.2 are the main issue of §5. Firstly, we give

examples of a C.C.CA. and a CA.F. Then we show the intimate connection of

Conjecture 5.2 with higher dimensional disconjugacy theory. Finally, we intro-

duce the higher dimensional generalization of nilpart and then extend the

results of §1 f rom single operators to families.

The prime examples of a CA.F. and of a C.C.CA. are constructed as fol-

lows. Hencefor th they will be called the paradigms for a CA.F. and C.C.CA.

respectively. Let D be a domain in R k with smooth boundary and define the

space K = L 2 ( k + I , D , A d ) to be IU+l-valued functions on D with

(5.6) Ilfl[ = fD (Af, f) [,k+l dQ, < co

Here A k = {aij}ij= 0 is a positive-definite matr ix-valued function on D and dE

is Lebesgue vo lume measure on D.

Now we construct a CA.F. as well as a J.F. which extends it. On K

define Jordan operators J% for ,~ = 1 ..... k by JR = m×9~+ N~ where Mx~ is

multiplication by the ~,-th coordinate function and N~f = g is the Iff+l-valued

function whose u-th component g, is

(v = 0,1 ..... k ) . gv = f0 (the 0 - c o m p o n e n t of f) if v =

Ball and Helton 276

Clearly NkN~, = 0 and the family J satisfies (5.3). An invariant subspace is

H = {u = [u,u× 1 ..... Uxk] t E K:

u absolutely cont inuous in each

variable, U~D = 0} .

0 Here Ux¢ = /)x--- 7 u. The action of J¢ on u in H is

J~,u = [x~u,X~Uxl ..... X~Ux + u ..... X~Uxk ]t

a (xu) a a = [x~u' ~x~ . . . . • ax~ ( x u ) , ' ' aXk (x~ %)1~

which is again in /-L Note that H is isometrically equal to a type of Sobolev

-space H on D, namely

k

{u,ux~ . . . . . Uxk ll ~ H ' ' (u ¢ H with ]lul~ = ~ JDD aijUxjUxid~'} " i,j=0

where u×0 means u. If we set T~ = J~ IH, then T~, is unitarily equivalent to

Mx~?, on the space H, and T consti tutes a CA.F. Clearly J is a J.F. which

extends T.

One can construct a C.C.CA. (which is not a CA.F) and a C.C.J. extend-

ing it by a variation on the construction in the previous paragraph. For clarity

let us restrict generality. Take K = L2(4,D,Ad~) where a00 = q,

a~ = a22 = a33 = 1, and aij = 0 if i ;~ j. Define two Jordan operators

J l = M × ~ + N l a n d J 2 = M x 2 + N 2 w h e r e Ni and N2 are the nilpotent opera-

tors of (5.4). Then J = {J1,J2} is a C.C.J. which is not a J.F. An invariant

subspace for J is H = {u = [u,ux~,Ux2,Uxv~2] t E K: u is absolutely

Ball and Helton 277

continuous and U~D = 0}.

Here Uxi = 0 0 2

¢3xi u, and uxl×2- 8xlOx:

H is

u. Indeed the action of J~ on u in

JlU = [xlu, XlUx I + U, XlUx2,XlUxlx2 + Ux2]t

--- [xlu, (xlu), (xlu), axlOx2 (xlu)]'

which is again in H, and similarly

a (x2u) 0 0~ J : , = tx :u , , ( x : u ) l , .

The operators T1 = JI[H and T2 = J2lHform a C.C.CA. which is not a CA.F.

We have been assuming that A gives a Hermitean form by (5.6) which

is positive definite on all of K. More generally assume only that the Hermi-

tean form is positive definite on H. As we saw in §1 when k=l this can hap-

pen without A being pointwise positive definite. However, T as defined in

the previous two examples is a CA.F. and a C.C.CA, respectively. Since the A

Hermitean form on K is not positive definite the J which were constructed

are not Jordan extensions of T. Consequently the existence of a Jordan exten-

sion is problematical. This example is much more indicative of the general

situation than one would think. The reason is that any C.C.CA. with a cyclic

vector can be represented as {M× }k on a type of Sobolev space over some

domain D. This is described in a rather crude theorem of [H3]; the theorem

does not determine the order of the Sobolev space nor distinguish the case

where (5.5) holds from the general case. We now analyze J.F. extensions of

our paradigm carefully.

Ball and Helton 278

T H E O R E M 5.4. Let H be the Sobolev space of functions on D which van-

ish on OD with inner product

(f,g> = f {fxlgx I q- - - --1- fxkgx k q- qfg}d,~ @

D

Here we assume that (.,.> is positive definite and that q is once differentiable.

Then the operators Mx~ . . . . . Mxk on H extend to a commuting Jordan fam-

ily J (i.e., (5.3) holds).

Proof. We shall ultimately construct a J which extends T to ~ space K

with precisely the form given in the preceding example (from equation (5.6)

on). The key is to find the right positive definite matrix-valued function A on

D in equation (5.6). The key property of A is

(5.6') ( f , f ) = IlflL for all f in H

since if this holds one can simply select J as in the example and thereby have

a J.F. which extends Mx. One Hermitean form which satisfies (5.6') is , 0

where Q = {qij} is defined by

q011 = q, qi j = 1 if j ~ 0, and q i j = 0 if i ~ j

Unfortunately, /x , }O is positive definite on K only if q is a positive definite

function. The follOwing lemma classifies all A which satisfy (5.6').

LEMMA 5.5. The sesquilinear form (',-}A with differentiable A equals the

Ball and Helton 279

sequilinear form <.,.)O on H i f and only if

A =

"q+BI +B2 +... +B k 61 ~2 , x i z'- 2 ~

- # 1 I iYl ,2

#~ -±YI ,2 i

° ° ° ° ° . , ° . °

B k -iy~,k -zY2,k

m

k ,q

rYl ,k

iY2 , k

1

where yij (1 ~< i < j ~< k) are real-valued differentiable functions, and each

/Y is a function satisfying

Proof. Let B = A - Q and suppose

u

Ux 1

U x k

v]> V x 1

V x k C+,

d = 0

for all u,v in C~°(D). This is equivalent to

F*BFu = 0

for all u 6 C~ °, where

lvl 1+h Fv___a v×, = ,J

1

Ball and Helton 280

and naturally the adjoint of F with respect to f ( - , ->~ on C~ ° is D

[° F * = 1, O X 1 . . . . . .

If we write B as {bij} k, we then get the equations

k O k O O O=Eboj u-E bij j=O i=l ~ x j

j=O

where ~ u = Uxo means u. This in turn becomes

0 = boo - bio)xi u + boj - bjo - bij)x i uxj

k

-- E bijuxixj " i=l j=l

Since this holds for all u in Cg ' (D) , we can conclude successively

boo- ~ (bio)~ i = 0 i=l

boj - bjo - E (bij)xi = 0 i=l

and

bij + bji = 0 if i ~ 0 and j ~ 0 .

Since bij = bji , we conclude that bij is imaginary for i ~ O, j ;e O, and bii = 0

f o r i ~ O. SetB j = b o j , T i j = i b O (1 ~< i < j ~ k), and the lemma follows.

Ball and Helton 281

We next seek a positive definite A of the form in Lemma 5.5

corresponding to the special case where all y~j = 0; then /3J is real. Positive-

definiteness of A is equivalent to the positivity of the principle minors of A.

The principle minors are

k where Z represents ]~/3{;

j=l

one, det A, is positive.

q + 7- -- (/31) 2

q + E -- (/31)2 _ (/32)2

k q + Z - Z ( / 3 J ) 2 = d e t A

j=l

Clearly they are all positive if and only if the last

Since the Q-inner product is positive definite on /4,

small, the Qo-inner product is also, where Q0 = {qi 0} with

for ~ sufficiently

qo°o = qoo - ~ and

qi ° = qij when i,j ;~ 0 (see inequality 1.6.3 of [M]). Set A0 = Q o + B; if

d e t A o = 0, then d e t A > ~, and hence A = Q + B is pointwise positive-

, definite. Now the equation det A0 = 0 is quadratic in /3J and linear in their

derivatives, i.e., it is Riccati-like. This suggests the substitution

/3J = - - Uxj /U ;

and one computes formally

det A0 = q0 Au Lou

U U

where L0u = - Au + qoU. By the discussion in §1.6 of Morrey 's book [M],

since Q0 is positive-definite on H, there exists a solution u of

Ball and Helton 282

Lou = 0

which is positive throughout the closure of D. Thus each /3J is a C 1 function

in D with no singularities and from these we can construct A0 and hence an

A having the desired properties. That proves Theorem 5.4. Note the construc-

tion is possible if a non-vanishingsolution u to L0 u = 0 exists. Conveniently,

if every solution v has a vanishing set equal to a smooth hypersurface

inside D which splits D as in the figure, then the function

v 0 ~ { O inside ~

outside /~

lies in H but satisfies fDL0V0~0 = 0 which implies v0,v0 < 0, a contradic-

tion. This type of analysis is typical of higher dimensional conjugate surface

theory (cf. [M]) and clearly is also closely connected to our extension theory.

- 3D

Figure.

It seems plausible that an analogue of Theorem 5.4 is true for our para-

digm for a C.C.CA. The case above (i.e., the paradigm for a CA.F.), restricted

to k = 2, involves factoring the second order positive-definite differential

02 O~ operator L0 0x~ 0x~ + q0 as

Ball and Helton 283

L 0 = 0 01 l l

8xl 8x2

where A is a positive-definite matrix of functions. To handle the paradigm

for a C.C.CA., we must factor the fourth order (but only of order two in a

0 2 8__L given variable) positive-definite differential operator L0 = 8x~ 8x~ +

8 4 + q0 as

Ox(Sx

l 8 8 8 2 Lo = 8xl t3x2 8xtSx2

8 8xl

I ° - - A 8x2

8 2

8xl8:

This requires a conjugate point theory more general than that discussed in

Morrey's book [M].

So far we have generalizedthe motivating Sobolev space example in § 1

to several variables in two settings (CA.F. and C.C.CF). Next we develop the

analogue (first for CA.F. (Theorem 5.6), then for C.C.CF. (Theorem 5.7)) of

the more abstract Theorem 1.1, which sets up a one-to-one correspondence

between Jordan extensions and nilparts. We also sketch a possible connection

between higher dimensional nilparts and factorization of the symbol expansion

QT(S), a possible several variable extension of § 2. The details of such a gen-

eralization however look formidable and we do not pursue them here.

Ball and Helton 284

D E F I N I T I O N . Given T a CA.F. on H with symbol expansion

k k

Qv(s) = I + ~ , sflj + ~ , sisflij 1 ij=l

the family A of operators on H is said to be a nilpart for T if fA ]. (5.7) [Al ' Ak] ~< B

[a;]

where the matrix B = {Bij}i~=l has entries

(Aj + A/)Ti}

(5.8)

(5.9)

g i j = 1/) /~ij -["

A , - Aj = Vj - Vj = - i ¢ j

Ti (Aj+ A/) - (Aj+ Aj )Ti

+ Tj*(Ai+Ai*) - ( A i + A i ) T j = 0 .

'/2 {Ti*(Aj+ Aj*) -

A is a strong nilpart for T if in addition

(5.10) for each there is a k so that

<f,O ~ k {llfll2-11wfllq.

Here

and B'/:W =

k W: H---, ~1 H is the unique contraction operator with ran W C [ran B]-

1

• Note Bij = CTi°C-rj(I). We note for future reference that an

l<1

Ball and Helton 285

equivalent definition of Bij in (5.7), given that (5.8) holds, is

Bij = Ti*Aj - AjTi.

Our first generalization of Theorem 1.1 is

T H E O R E M 5.6. If the coadjoint family T on Hextends to a Jordan family

J = S + N , then A = P H N I H is a strong nilpart for T. Conversely, if T i sa

coadjoint family and A is a strong nilpart for T, then T can be extended by a

Jordan family J with A = PHNIH. Moreover, A determines a minimal

extension J uniquely.

Proof. Suppose J is a J.F. extending T and A = PHN IH. Then

Bij = Ti*Aj - AjTi = PH{ (Si + Ni*) PHNj - Nj (S i q- N i) }[H

-- PH{(Si+N i )Nj - Nj(Si+ Ni)}~H

= PHNi Nj [H, since J satisfies (5.3) .

Thus Bij + Bji = PH(Ni*Nj + Nj*Ni)IH = CL° CT(I) = fl0 by direct computa-

tion (again using (5.3)), and (5.9) follows.

B = P k ~ H I

Since Bij = PHNi NjIH, if B is the block matrix B = [B0], then

• k

[ N I ' ' ' N k ] I ~ H while I

IA I 1 [ A I ' " ' A k ] = P ; H

lAq P H [ N j ' ' ' N k ]

Ball and Helton 286

and (5.7) follows. (5.8) follows as in Theorem 1.1, since in particular Aj is a k

nilpart for Tj. Finally, if we let A = [A1 " ' " Ak]" ~ H---' K be defined as I

k [Ni "" " S k ] l ~ H, then B = A'A, and hence A has polar decomposition of

k the form A = W~B ~/2 where W~P~:@ H---. K is a partial isometry with initial

l

space [ran B]- and final space [ran A]-. Then it can be checked that

W = W I I H (W as in 5.10). Hence, for f 6 /4,

Ilwfl[ x = IIw,f[F < IIPR~=tN, ... N~lflF

-< II Pkor N~fTF

(since Ran[N1 ' ' ' Nk]-- V { R a n N " =1 ..... s} c kerNj by (5.3)). Hence,

for any f in H,

CT2j(1)f, f H = IINjfIF = IINjP~k~N? t]F

~< ][Njlff {]lflff -IlPkerNj fl~}

< IINjlI2{IIflF -Ilwfl~}

and (5.10) follows.

Conversely, suppose T is a CA.F. and A satisfies (5.7-10). Let K1 be k

the completion of ~ H in the inner product (f,g>K~ = (Bf, g)k , and let ~ H

1

k i: ~ H ---, K1 be the inclusion map. We verify the identity

I

[+l{+ I (5.11) B T = T* B or BijT~=T~ Bij

Ball and Helton 287

as follows:

(T i A j - A j T i ) T ~ - - T~ (Ti Aj- AjTi)

I -T; T( %+% 2

(using (5.8) and CT i o CTj

= [T~* % + 2 %*

1 %+% ~r~ 2

° CTg (I) .= 0)

- i = T {CT¢ (Bij) + CTB (Bji)}

( u s i n g Bij = Bji , since B is positive definite by (5.7))

i {CT£ (Bij -t- Bji) } -

i o = - ~ C-r~, CT~ ° C-rj(I) by (5.9)

A j+%* I TiIT~

= 0 by ( 5 . 4 ) .

Since T is a CA.F. it has a functional calculus; f (T 1 ..... Tk) 1 (2 I r ) kn

f f ( x l , . . . , x s ) • • • dx l • • • dXk. T h e above iden t i t y (5.11) imp l ies eiXlTl eiXkTk

that there are k commuting self-adjoint operators SI ..... Sk on K2 such that

f(S11 .... ,Skl)i = i f Tl . . . . . ~ T k for f 6 S(IR k). In particular, for 1

suppf N sp(T1) x . - . x sp(Tk) = 4,, we have f(S~ . . . . . skl)i = 0. Since

i has dense range, f(S~ ..... S~) = 0. Since S~,...,S~ are commuting self-adjoint

operators, by the spectral representation theorem, this implies that I[s~ll < IIT~ II (g = 1 ..... k).

Ball and Helton 288

Next we define an operator D: H---* K1

k

<f,[Al ..... Ak]g>H for f C H a n d g E ~ H. Then 1

(5.12)

\ [ A q /

[ k

~ H 1

by (Df, ig>K 1 =

= ]<Wf, B'/~'g>k I < [[Wfllk IlB'/2gllk ~ H ~ H ~ H

1 1 I

< [Iwl]l]fl[H IliglL%

and hence D is bounded by ]lW]l.

[ [A ll [Aql

(5.13) DT~, = S~D + ig

[+1 where i~: H---, Kl is defined as i~(f) = i fj

k k

for f6 H , g = ~ g n C~H,1

Next we verify the identity:

{ f, j=~, Indeed where fj = 0, j -~

<DT~ f, ig>,v, = <T~, f,A,gl +.-.+ Akgk )H

Ball and Helton 289

-- <f, (B~,l + A1T~ )gl + " • • + (BZk + AkTL)gk}H

by definition of B£n

= < f , AITLgl+ " ' ' +AkT£gk>H+<f ,B£1g l+ ' ' " +B£kgk>H

I+1 = <Df, i T~, g>K2+<i~f ,g>K 2

= ( (S~D+i~) f , i g)K2.

We next define

norm

Ko to be the completion of H in the Hilbert space

Ilfll~ = Ilql 2 - I l W f [ l ~

and denote by io: H - - , Ko the inclusion map.

I,o 1 map 1: f - - . [Df] ' We note

sup {l<Df, ig>K.l" Iligl[K, ~ 1}

,/, = sup{]<Wf, B-g>l.llB'/-'gllk ~< 1} by (5.12)

~ H I

= IlWfll, or for all f ¢ H,

Let 1: H- - - K o ~ Kj be the

IIDfIIK, = IlWfllH •

Hence the map I is isometric. We next check that, for ~, = 1,...,k

(5.14) (1-W*W)T¢ = "1-~ (1 -W'W) .

Ball and Helton 290

The argument is formal in that it involves B -1, and B may not be boundedly

invertible even when it is injective. However, this can be remedied by inter-

preting B -l as the standard generalized inverse, and checking that the projec-

tions arising in the computation with a generalized inverse do not obstruct the

validity of the equations. First, from (~T~)B= B(~T~), we conclude

B-' (~T~) = (~T~,)B-'. Thus W*WT z - T~, W*W

~,A, ~,~'1 i I~ ~ a, AjB-lJi j [Aq [Aq

= [ A ' " ' " Ak]B-I] ! ] - T * [ A ' " ' " A J B - ' i,

[B~ + T~ A'k] lAkl

by definition of B~. This in turn equals

= [ A I " ' " AklB-1BF~+[A' " ' ' AklB-I(~T.~ )1 .:,1

lAq

JAil - T~, [Al • ' • Ak]B -1

lAq

where F_g is a column operator matrix with k components, the ~th_

component equal to the identity I, the others zero. This then equals

Ball and Helton 291

= A~ + [AIT£-T~AI - - ' AkT~.

= A z+[-Bz~ ' -

akJB' I i1 IAq

IA~I lAkJ

=A£ - A ~ =T~,-T~ by (5.8),

and (5.14) follows. In the same way as above, this gives commuting self-

adjoint operators SI ~ . . . . . S o on K0, with IlS~ll ~< [ITz[I (z = 1 ..... k) such that

i0T z = S~°io.

Finally, we define operators N~: Ko---" K~ by N~: iof--" izf for f 6 H

and ~ = 1 ..... k. Condition (5.10) implies that this formula extends to all of K0

by continuity to define a bounded operator: Let us define J£ on K = K0 • Ka

js 01 i001 byJz = ( = S ~ + N ~ whereS~ = and N ~ = ). Then IN~S~ s~ N z 0

it is easily checked that J is a commutative family of Jordan operators

Ball and Helton 292

satisfying (5.3). Also, for f 6 H,

S D]

= ]ioTa] = IT£. [DT~]

Hence, if H is identified with its isometric image in K via L then J2H c H

and J~]H = T~. The uniqueness assertion of the theorem follows as in the

proof of Theorem 1.4. Theorem 5.7 now is proved.

It is interesting to compare the abstract construction in the proof of

Theorem 5.6 with the concrete construction in the proof of Theorem 5.4. It

can be shown that assuming all 7 's equal zero in (5.5) amounts to assuming

Bij = 72 CT~ ° CTJl), or equivalently

(5.9') Ti*(Aj+ Aj*)= (Aj+Aj*)Ti.

In the context of Theorem 5.4

D: H----, K~ is equal to

the space K1

d

st d

of Theorem 5.6 is L2(k,D,d£),

is multiplication by x~,

K0 = L2(1,D,(q-q0)d£) with i0: H--~ K0 being inclusion, S~ also is multipli-

cation by x~, and {ill . . . . . /~k} satisfying the Riccati equation detAo = 0 is

equivalent to D*D = q o - A (where * is the adjoint with respect to the

L2(k,D,dO inner product). Whether it is always possible to obtain a Jordan

lift J such that the corresponding set of nilparts A satisfies (5.9') rather than

the weaker (5.9) is not clear.

Ball and Helton 293

Also there is a

M(S) = 1 + s i a l + " ' ' + SkAk.

(5.8) together with

connection with factorization. Let

Then M(S)*M(S) = QT(S) is equivalent to

{Ai Aj + Aj*Aili~j=l = ]3ij = Bij + Bij .

Thus, if it happens that A = {A1 . . . . . Ak} also satisfies (5.9'), then (5.7) is

satisfied with equality and A is a nilpart for T. If it happens that A satisfies

only (5.9), then a symmetrized version of (5.7) is satisfied with equality.

Finally, we extend Theorem 1.1 to a C.C.CA. First we need some nota-

tion. Let T be a C.C.CA. (that is, T is still commutat ive but, may not satisfy

the stronger condition (5.4) or (5.4')) on H and suppose J is a C.C.J.

extending T (so by Proposition 5.1, J satisfies (5.2)). For a k-tuNe

= (gl . . . . . gk) consisting of 0's and l ' s with at least one 1, let N ~ be the

product N~ '~ ~'k = • ' " N k , and set A, PHN"IH, and B~,,= PnN*~'N"I/-/. Then

we observe

IA l 1

[A1 ' ' " A ~ I = P ; H

/

PH[N ~'' . . . N.~I [. ~ B ~ H 1

where K = 2 k - 1 = the number of such k-tuples /x and B is the K by K

matrix, rows and columns indexed by k-tuples /x, with components B, ,

defined as above. For p any k-tuple of nonnegative integers, we define

= I . . . . .

where CT~:X ---~ i ( X T ~ - T ~ X ) , and S~ '= s~ 'l - . . s~ k and p ! =

Pl!P2! ' ' Pk! , lO[= P ~ + P 2 + ' +pk, and [p~ ,= m a x { p l , . . . , p k } . Then

Ball and Helton 294

the symbol expansion for T can be written compactly as

QT(S )_= y . ~S,,sV ' = ~ 1 [t,~ <2 I,,k~-~ 2 ~ C~ (1)sV

= 7_,{ 7_ p p. ,v

tx+u=p

Here (i = .,/-2-f and # + t, is defined componentwise). Hence

(5.16) ]~ (-j)lui(i)l"JB.,, =/3,, p..V

# + v = p

Also it is easy to check

i CT~ (Bu,~) = {T~*PHN*UN ~ - PHN**'N'T~}I H

= PH{ (S~ + N~) N * " N ~ - N*~'N ~ (S~ + N~)}I H

= pH{N*~+O~,N , - N*gN~+%}I H

= B ,+%. , - S,~.~+%.

Here e~ is the k-tuple with all 0's except for a 1 in the ~,-th slot.

motivates the following.

DEFINITION. Given

with symbol expansion

This

T a family of commuting coadjoint operators on H

QT(S) = ~ /3vs v,

the family A = {A~,: I/a,~ ~ 1} of operators on H is said to be a nilpartfor T

if

Ball and Helton 295

(a) the inequality

A I

(5.15) [A~ - . - AuK] ~< B = [B...I

lA: J

holds, where Bu,. is determined by the recurrence relation

(5.16) B.+o ,. = iCT£(B.,.) + B;,,.+e£

together with initial conditions

B..o = A., Bo,. = A . ,

and

(b) for all k-tuples P, loL ~< 2,

(-i)bl(iff t B~.. =/3 o . /.¢,,v

# + u = p

The conditions for boundedness are less pleasant. We say that

strong nilpart for T if, in addition to the above,

(c) For all k-tuNes /x, [ ~ ~< 1, and £ = 1 ..... k,

(5.17) /321u+e£ ) ~< k£/32;~ (multiplication of k-tuples

by a scalar also is defined componentwise)

and also,

- * BI~ B-Y2W (5.18) Be£,e £ - W*B-Y"Fg*BEe Ee£

A .is a

Ball and Helton 296

ur*~-'/~r*~: Ee~ BF~ B-' :W (5.18) B%,~ff ,, ~, , ~ , , , , ~ - *

+ B '/2W k z (1-W'W)

where F;~ is the shift operator {hu} ---, {hu_e~} on ~ H, Ee ~ is the column /z

luL~<l

matrix with all components 0 except for an identity I in the e~ slot, and

His the unique contraction operator with W: H--"

A i

ran W c [ran B]- and B'/-~W =

lA q (It can be shown that if (5.17) holds, all terms in (5.18) are bounded, despite

the appearance of the formal inverses.)

Now we can state our ultimate theorem.

T H E O R E M 5.7. If the commuting collection T of coadjoint operators on H

extends to a commuting collection J = s + N of Jordan operators on K, then

A = {A, = PHN"IH} is a strong nilpart for T. Conversely, if T is a C.C.CA.

and A is a strong nilpart for T, then T can be extended to a C.C.J. J with

A, = PHN"{H. Moreover A determines a minimal extension J uniquely in

this way.

Proof. As the reader can imagine, the bookkeeping involved for a full

rigorous proof is greater than what has come before by an order of magnitude.

We merely sketch the construction with the assumption that T has a Jordan

extension J, and A, = PHNU]H. Specialization to k = 1 provides a more

algebraic motivation for our original proof of Theorem 1.1. Let K1 be the

space ~ H with the B inner product, so

Ball and Helton 297

<@f.,@gu>A. = ~ <N*UN"f~,g.)

= < ZN"f~ , ]~ NUgu) . u #

g

Hence N"~f~+ ' + N"~f~--' j--~l fj extends by continuity to define an

isometry of R = (ran[N "~ . • • Nu~][ ~ H) - onto KI. Also we can show that 1

for f 6 H, IlWfll 2 = IlpRflff and hence, if we let K0 be the completion of H

in the inner product IIfll~0 = ((l-W*W)f,f>tt , then PRkf---' f is an isometry of

(pRI-H) - onto K0. By minimality of the Jordan extension J, P~ H + R is

dense in K, and hence

defines a unitary mapping of K onto K ' = K0 • Kl. To compute a formula

for UI H, we define the operator D: H --" Kl by

<Df, ig>,~., = <f,[Al " ' " A~]g>H

and check (at least formally) that

/IA l ,AfA ]i

Ball and Helton 298

[fl so, I: f---. is an isometric embedding of H onto K', and in fact agrees

D

with U on H. One then checks that

UJ

and

IT 01 [SO l = U and US = E -F D ((~T1) ( ~ T ) - V U ,

[0 0] UN = U .

E - F D F

(The operator matrices are with respect to the decomposition of K ' as

K1 • K2). The idea of the proof of Theorem 5.7 then is to use these operator

matrices to define operators J', S', and N' on K ' such that J ' = {J'} is the

desired Jordan extension of T ' = ITI*]IH.

§6. Complex Jordan Operators.

We say that an operator J on K is complex Jordan (of order n) if J has

the form J = M + N w h e r e M is normal, M N = N M a n d N n = 0 . We say

that an operator T on H is sub-complex Jordan (abbreviated sub-CJ) if there

is a Hilbert space K D H, a complex Jordan operator J on K such that

J H C H and T = J]H. We note that a sub-CJ of order 1 is called subnormal

in the literature and has been extensively studied; the general class introduced

here should provide a fertile area in which to seek generalizations of known

results on subnormal operators. If J = M + N is complex Jordan and M has

M+M* ,H 2 M-M* ], then by Cartesian decomposition M = H 1 + i H 2 H1 2 2i

Fuglede's theorem both H~ and H 2 commute with N. Hence J = J~ + i J2

Ball and Helton 299

(Jl = H1 + N,J2 = H2) where Jl and J2 are commuting (real) Jordan opera-

tors. The big distinction is that the subspace K need not be invariant under Ji

and J2 as it was in §5. So unlike the real case if T is sub-complex Jordan,

there may be no decomposition T = T~ + iT2 into commuting sub-real Jordan

operators T~ and T2. Also, unlike the §5 cases where we at least have plausi-

ble conjectures and unlike the case of a subnormal operator which has been

solved (see [Br]), we have no idea of how to solve

Problem 6.1. Give a simple condition on an operator T to determine if it

is sub-complex Jordan.

Even an algebraic characterization of complex Jordan operators analogous

to Theorem A [H1] isn't known. There is, however, an obvious way to start.

If J = M + N is complex Jordan with N 2 = 0, then C3 2

[[j , j ,] , j]2 = (NN,N)2 = 0. Is the converse true or are further algebraic rela-

tionships necessary? Note given

should be N by

J with C] = 0 one can actually build what

N ~ C3(C3C;) 1/3 ;

also a glance at null spaces implies N so defined is nilpotent. It still remains to

show that J - N is normal, etc.

Sub-~J operators have a paradigm similar to that for the real case. We

shall describe the order 2 case. Let D c t C= {xl+ix2},A, [][k and

K = L2(3,D,A) be as before (see (5.5)). However, here we set J = Jl + i J2.

There are several subspaces of K which are invariant under J. The subspace

H in the paradigm in §5 is certainly one in which case T=J]n decomposes as

T1 + iT2 where {T1,T2} is a C.CA.F. A radically different invariant subspace

is

Ou . O u ] , HA = {[u, --~-z ' 1 0 z E K: u analytic in D}

Ball and Helton 300

which is a subspace of H and J acts on HA via

iu0 u, ul ' ~ Z u ,

Naturally, the last entry in [u,uz,iUz] is redundant and an operator unitarily

equivalent to JluA can be described more simply (this approach suppresses a

certain unity between §5 and §6). At any rate the paradigm for sub-CJ opera-

tors is M~ on a generalized Sobolev space of analytic functions on D.

Cowen and Douglas [C-D] have recently studied another class of opera-

tors BI(D) associated with a domain D in the complex plane. We give an

equivalent definition which is more convenient for our purposes. An operator

T on a Hilbert space H is in BI(D)* (the set of adjoints of operators in the

Cowen-Douglas class BI(D*) where D * = {z:2 E D}) if

(a) k e r ( T - w ) = {0} for each w in D,

(b) r a n ( T - w ) is closed for each w in D,

(c) dim ke r (T*- ,~ ) = 1 for each w in D,

(d) M r a n ( T - w ) = {0}. wED

Douglas and Cowen observe that an operator T in BI(D)* is unitarily

equivalent to multiplication by z on a Hilbert space H of analytic functions

on D with reproducing kernel (abbreviated R.K.H. space). This means that

for each f in H and w in D, f is analytic on D and the map

f E H ---, f(w) E Cis continuous; hence there is a kernel function k(w,z) such

that k(w,.) E H whenever w E D, and <f,k(w,.)) H = f(w) for all f in H. We

show that the converse direction follows under mild additional hypotheses.

P R O P O S I T I O N 6.2. Suppose T is multiplication by z on a R.K.H. space H

of analytic functions on D and 1 E H. Suppose for each w E D,

Ball and Helton 301

f ( z ) - f (w) H whenever f(z) 6 H. Then T is in B~(D)*. In particular if

Z - - W

H = HA of our prime example and Ilfl~A >i 8 f Ilfll2d for all f, then the ~O

sub-Gl operator T is in Bl(D)*.

Proof. If f E ker (T-w) for some w 6 D, then (z-w)f(z) ------ 0 on D,

hence f--- 0 and (a) holds. By assumption, any f(z) in H can be written as

f(z) = f(w)l + ( z - w ) f ( z ) - f ( w ) where f(w).l and f ( z ) - f (w) are in H. Z - - W Z - - W

Hence ran(T-w) has codimension 1, and (c) follows. Also

ran(T-w) = {f 6 H: f(w) = 0}. Since point evaluations are continuous in H

norm, we conclude ran(T-w) is closed, and (b) follows. Finally, from the

above, if f is in N ran(T-w) , then f ( w ) = 0 for all w in D, and hence wCD

f---0. Thus T is inBl(D)*.

To conclude the proof we must only check that HA is a R.K.H. space.

By the Mean Value Theorem for harmonic functions, for f E HA, and w in

D, f(w) is equal to the average value of f over any small disk A centered at

w. So

If(w)[ ~< (AreaA)-IfD Ifld

~< (&AreaA) -1 Ilf[hzA

and so evaluation at w is continuous in HA-norm as required.

We next wish to contrast the situation for HA to that for a "real" Sobolev

space based on a domain D C R k. For simplicity we consider only Sobolev

spaces of order 1.

PROPOSITION 6.3. Let H be the Sobolev space with inner product ( , >O

as in Theorem 5.4 over D C R k.

Ball and Helton 3,02

(1) For k = l , H is a reproducing kernel Hilbert space, and

M x - x for X in D is n o t F r e d h o l m .

(2) For k >/. 2, H is not a reproducing kernel I-Iilbert space.

Proof. For k = 1, a Sobolev l emma (Theorem 3.5.i in [M]) guarantees

that the e lements of H are cont inuous and that point evaluations are continu-

ous on H. HenCe H is a reproducing kernel Hilbert space. Also, it is easily

seen that k e r ( M ~ - X) = 0 and [ r a n ( M × - ~ . ) ] - = {f E H: f(~.) = 0} for

X E D. If it were the case that M x - X were Fredholm, then

c o d i m [ r a n ( M ~ - ~ . ) n] = _ i n d ( M × - X ) n

= - n i n d ( M x - X ) = n ,

while, in fact, it is easily seen that [ r a n ( M ~ - X ) n ] - = [ r a n ( M × - X ) ] - has

codimension equal to one. Hence M × - X cannot be Fredholm. This illus-

trates that some hypothesis in addition to H being a reproducing kernel Hil-

bert space in Proposition 6.2 is essential.

For k >/ 2, it is easy to prove that point evaluations are not continuous.

For example , for k = 2, consider a function f whose Fourier t ransform is

cont inuous and which near oo satisfies

~({:) = ~ log I~:l

Now f is in the first Sobolev space because (1 +[~:]2) ]~(~.)]2 is integrable,

1 f ~(~)d~: = oo So evaluation at 0 is not a cont inuous func- but f(0) = ~

tional. This example came to the authors f rom P. Gilkey through J. V. Rals-

ton.

We next turn our attention to the converse of Proposition 6.2.

Ball and Helton 303

Question 6.4. Which operators T in Bl(D)* have a complex Jordan

extension?

We henceforth consider a T in BI(D)* of the form Mz on a R.K.H.

space H of analytic functions on D. Note that when H = HA in our para-

digm the inner product < , ) on K extends to a "local" inner product < , )A

on C3(D) C K. Here a local form ( , ) is one which satisfies ( f , g )= 0

whenever f,g in C3(D) have disjoint support. The spectral representation for

a complex Jordan operator J makes it clear that Question 6.4 for a sub-CJ

operator T is closely related to

Question 6.5. Which positive definite Hermitean forms < , ) on a

R.K.H. space H of analytic functions on D have a positive definite local

extension to C3(D)?

In a moment we shall describe a form ( . , . ) on H with no such local

extension. In the process we shall give

Example 6.6. An operator T in BI(D)* with no Jordan extension of any order.

Let D be the unit disk, define a Hermitean form on polynomials p,q by

2rr

(p ,q ) = f p(e i°) q(e ~) dO + [p(1) + p(-1)] [q(1) + q ( -1 ) ] .%

and let H be the completion in < , > inner product. Then H satisfies the

hypotheses of Proposition 6.2 so T = Mz is in BI(D)*. While a locality test

makes no sense for a space of analytic functions, we shall show that ( , )

actually violates the following "asymptotic locality" condition which must hold if

T is sub-CJ. Consider two sequences

fn z [l*Zl a n

Ball and Helton 304

which asymptotically have disjoint influence on D-; that is, fn~J)gnn ~£) "--" 0 uni-

formly on D- as n ~ oo for all j,z = 0,1 ..... One computes from the

definition of ( , ) that <fn,gn> "-" 1 as n---, oo. Hence ( , ) cannot have a

local extension to CqD) . To see that this condition is inconsistent with T

having any CA extension of any order (including oo), we need the following

general fact. The proof will follow after the discussion of the example is com-

pleted.

LEMMA 6.7. If T is sub-cA (of any order) on H with minimal CA-

extension J, then sp(J) C sp(T).

Now assume the lemma and suppose that T has a CA extension

J = M + N on K where M is normal with spectral measure E(dh), N is

quasi-nilpotent and M and N commute. By the lemma E(dh) i s supported

on D-. Then

Kfn,gn)l = I(fn(T)1,gn (T)1>[ = [(f,(J) 1,g.(J) 1)h~

N}n Ninl) = I + E ( d h ) l , I - T I

D -

~< islu~tl ] + II II 12"-----~h I - II 111112.

Now note that D- C D~ U D 2 where D 1 =

D2= I~< - - . Choose p > 0 but sufficiently small so that - - + p

(l+p) < 1. Since ]/2N is quasi-nilpotent, there is a constant C such that

~ IINJII ~< Ct; for j = 0,1 Then for h E D- N D],

[_~_~_ ~1 n {~}{ i+h i lr II I + It = II ~ T 5 - II s=0

Ball and Helton 305

~< Ck~o [~ I ]._..~.__~<I÷A. pn-k

Similarly,

II l -

C(1 + p ) n

Since ( l + p ) < 1, we have

NIn,, sup I -

k E D - N D I

- - ' 0

as n - - ~ .

A similar argument works for D- n D2, and hence the above estimate

implies

<fn,gn) -'* 0 as n ~ oo ,

a contradiction. We conclude that T cannot have a CI extension of any

order.

Proof of Lemma 6.7. The argument is an adaptation of that of [Hal] for

the case where J is normal. Via translation it is enough to show that if T is

invertible then so is J. Let J = M + N on K where M is normal with spec-

tral measure E(d,~), as above. If 0 6 sp(J), it is possible to choose an e,

Ball and Helton 306

0 ~< • < 1, such that Eo = E({Izl ~< •}) is a nontrivial spectral projection. We

may also assume IIT-~II ~< 1 Then if k 6 EoK and h 6 H,

I(k,h)l = I(k,TnT-nh>l = IO*nk,T-nh)l

= IIJ*nkll Ilhll •

However IIJ*nkll = Ilfl'~lji (hi + N*)"E(dh)k][ ~< iSl~P II(x + N)~II Ilkll, This

uses N commutes with E(dk)which holds since N commutes with M.

Now choose p > 0 so that t9 + • < 1. Since N is quasi-nilpotent, there is a

constant C such that IINJl] ~< Cp i for all j = 0,1 ..... Then

[IJ*nk[I ~< sup II ~ [~1 kkNn-k[] I ~i~<' k=O

k=O

k=0

and hence llJ *n kll - - 0 as n - - oo. Therefore E0H is orthogona] to /-L. Since

J is .minimal, it now follows that E0 = 0 and hence J is invertible. The

lemma follows.

Finally, we conclude by mentioning a special type of complex Jordan

operator, namely, a J whose normal part is a unitary operator U. An invari-

ant subspace H for such a J gives rise to an operator T = JIH. These opera-

tors split into two cases:

Ball and Helton 307

(1) Bilateral, i) T" = JnlHfor n any integer.

(2) Unilatera~ i fT n = JnlHholds only for n >/ 0.

Naturally, bilateral extensions behave just like the "real" Jordan extensions stu-

died in Sections 1-4. Their spectra are contained in the unit circle and they

satisfy

T*nT n =I + n B1 +n2B2

for all integers n. A unilateral T satisfies this only for n >~ 0. In typical

examples the essential spectrum of T is contained in the unit circle.

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Department of Mathematics Virginia Tech Blacksburg, Virginia 24061

Department of Mathematics University of California, San Diego La Jolla, California 92093

Submitted: February i0, 1980