Mathematics. - On the theory of linear integral equations. VIII. By A. C. ZAANEN. (Communicated by Prof. W. VAN DER WOUDE.)

(Communicated at the meeting ol April 26, 1947.)

§ 1. Introduction.

This paper is an immediate sequel to the paper VII, bearing the same title I). For a general introduction into the theory of systems of Iinear integral equations with "symmetrisable matrix-kernels" and for the meaning of the used symbols and notations we refer to that paper, in particular to § 1. In the present paper a knowledge of the contents of the papers I. 11 and VII, which we re published in these Proceedings 2) under the same title, is assumed.

We shall discuss here the system of linear integral equations

2J=1 J Klj (x, y) fj (y) dy-).[i (x) = gi (x) (i= 1, ... , n), (1)

where fi (x) and gi (x) € L~m) (.6), ). is complex, and the matrix-kern el 11 Klj (x, y) 11 is the matrix-product of the matrix-kern el 11 Aij (x, y) 11 and the matrix 11 hij (y) 11, the following conditions being satisfied:

a. All hij (y) are bounded and measurable in .6,

b. hij(y)=hjdy) and 2~j=1 hij(y)aiaj-;;::'O for any system of complex numbers al, ... ,an and every y€.6: the matrix Ilhij(y)11 is therefore Hermitian and of positive type,

c. All Aij(x, y) € L~m)(.6),

d, Ilhij(x)II·IIKij(x,y)II=IIKij(x,y)II·llhij(y)ll,

in other words,

11 hij (x) 11 . 11 Aij (x, y) 11 . 11 hij (y) 11 = 11 hij (x) 11 . 11 Aij (x, y) 11 . 11 hij (y) 11·

This last condition is automatically satisfied if 11 Aij (x, y) 1I is Hermitian. As we have already remarked in VII, § 1. the system (1) reduces for

n = 1 to the integral equation which we have discussed in 111 3). It wil!, however, be possible to read the present paper without any knowledge of the contents of 111. The case n -;;::'1 was considered by J. ERNEST

1) A. C. ZAANEN, On the theory of linear integral equations VII, Proc. Kon. Ned. Akad. v. Wetenseh., Amsterdam, 50, 357-368 (1947) (Indagationes Mathematicae IX, ,.~).

2) A. C. ZAANEN, On the theory of linear integral equations, Proc. Kon. Ned. Akad. van Wetenseh., Amsterdam, 49, I, 194-204, II, 205-212 (1946) (Indagationes Mathe­maticae VIII, I. 91-101. II, 102~1O9).

3) A. C. ZAANEN, On the theory of Iinear integral equations lIl. Proc. Kon. Ned. Akad. v. Wetenseh .. Amsterdam. 49. 292-301 (1946) (Indagationes Mathematicae VIII, 161-170).

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WILKINS in 1911 4) (he generalized earlier results of W. T. REID S)). who. however. supposed. instead of condition a.. all hij (y) to be continuous on b... and. instead of condition c.. all Aij (x. y) to be bounded in b.. X b.. with their discontinuities "regularly distributed". We shall free ourselves of these restrictions. and. moreover. prove a great deal more.

Af ter some preHminary lemma's in § 2 (one of these. Lemma 1. is of ' interest in itself). we shall show in § 3. amongst other things. that the characteristic values of (I). if any exist. are real: that under very general conditions this system possesses at least one characteristic value '1=- O. and that the characteristic values have some remarkable extremizing proper ties. Furthermore we shall prove expansion theorems for functions

of the form Il=1 f K ij (x. y) fJ (y) dy. as weIl as for the elements of the IJ.

matrix-kernel 11 K iJ (x. y) 11 themselves. In § 1 we shall consider the iterated kern els of 11 Kij (x. y) 11. and in § 5 we shall show that. by strengthening the conditions a. and c.. the results of the expansion theorems can be improved (replacement of convergence in mean by uniform convergence). Finally. in § 6, we shall compare our results with those obtained by WILKINS.

§ 2. Some preliminary lemma' s.

Lemma 1. 1°. If IIh iJ11 (i.j=1. .... n) is an Hermitian matrix of

positive type (that is. hij =hJi and ~r.J=1 hij ai aj ~ 0 for any sequence al •...• an of complex numbers). there exists a uniquely determined matrix

11 MJ) 11. which is also Hermitian and of positive type. such that '

11 M'y') 11 . 11 M'I II = 11 hij 11·

2°. If now the numbers hij depend upon the parameter x. running through the interval b... and all hij (x) are bounded. measurable functions

of x for x € 6.. the functions h ~j') (x) are also bounded and measurable in b...

3°. Pinally. if the functions hij (x) are continuous in b... the same

holds for the functions h~j') (x).

Proof. 1°. Considering an n-dimensional unitary space with ortho­norm al base cpl ••••• CPn. the linear transformation A in this space.

determined by Acpi=Il=1 hiJcpj, is self-adjoint and of positive type. As well-known. th ere exist an orthonormal system 'PI •••.• 'Pn and non­negative numbers ÀI •••.• Àn such that A 'Pi = Ài 'Pi (i = 1 ....• n). The numbers Ài are therefore the characteristic va lues of A and the elements

4) J. ERNEST WILKINS, Definitely self-conjugate adjoint integral equations, Duke Mathematica! JournaIlI, 155-166 (1944).

ó) W . T. REID. Expansion problems associated with a system of linear integra! equations. Transactions of the American Math. Soc. 33. 475-485 (1931).

467

"PI are corresponding characteristic elements. Defining now the trans­

formation A'/. by A'/. "Pi = lï·"Pi. this transformation is clearly self­adjoint and of positive type. and satisfies (A'/'F = A. Moreover. A'/. is the only self-adjoint transformation of positive type. having this property.

Writing (A'/· fPi • fPj}--:- h~'~ (i.j= l. .... n). the matrix Ilh~~,)11 is therefore uniquely determined, Hermitian and of positive type. while moreover

so that

hij = (A fPi, fP j} = (A'/. fPi. A'/, fPj) =

~r=1 (A'/. fPi. fPk) (All. fP j. fPk) = 1:r=1 h~~,) h~~ •

11 h~)) 11 . 11 M~ 11 = 11 hij 11·

2°. IE "PI' •.•• "Pn are the characteristic elements of A. introduced in

1°.. the elements fPl ..... fPn can be represented by fPi = Ij=1 bij "P j.

where 1:1=1 I bij 12 = 1 (i = 1 •.•.• nl. Then it follows easily that for every integer 1?;:0 we have

n 1 --(A 1 fPi. fP j) = 1:k=1 (A 1 fPi. 1f!k) (fP j, "Pk) = 2f=1 lk bik b jk:

hence, if P(l) is an arbitrary polynomium.

(P(A)fPl. fPj)=2f=1 P(lk) bik bjk.

Furthermore. in the same way.

h(l/') (All. ) n 'I. b -b ij = fPi, fPj = 1:k=1 lk ik jk'

Let now all hij = (A fPi. fP j) be bounded and measurable functions of x on 6. Then the transformation A. depending on x. is also bounded. uniformly in x. The same holds therefore for the characteristic va lues of A. so that there exists an absolute constant M> 0, having the property that. for all x € 6. all characteristic values of A are contained in 0 ~ l ~ M. Let now e > 0 be given. Then there exists a polynomium P(l) such that IP(l)-l'/'I<E for O~l~M. Writing Pij=(P(A)fPi,fPj), we have therefore

uniformly for x € 6. This implies that, since the functions Pij (x) are

eVidently measurable on 6. the same holds for the functions h~y)(x). The boundedness of h~1)(x) follows immediately from

I h~J) (x) I = 12f=1 l'/c' bik bjk I ~ M'/, ~:=I I bik bjk I ~ M'/ •.

3°. If thefunctions hij (x) are continuous in 6. the same holds for

the functions Pij (x): hence. by (2). also for the functions h~~ (x).

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Before stating our Lemma 2. we recall that in I. § 5. we have considered symmetrisable transformations K in a HILBERT space R. having the form K = AH. where A is self-adjoint. H is self-adjoint and of positive type. and at least one of the transformations A and H is completely continuous. We have proved in that place an expansion theorem for elements of the form K f. another one for every solution f 'of K f - ,1. f = g. and finally th at the self-adjoint transformation

K = H'!, A H'!· has the same sequence of characteristic values *" 0 as K = A H. Here we shall show that these results are also valid in a slightly more general case.

Lemma 2. All results. proved in I. § 5. remain valid when A is not necessarily self-adjoint. but satis{ies HA H = HA * H.

Proof. (HK{.g) = (HAH f.g) = ({.HA* Hg) = ({.HAHg)=(f. HKg); the transformation H K is therefore self-adjoint. in other words. K is symmetrisable (relative to H). The proofs of the expansion theorems in I. § 5. remain unaltered; furthermore it is easily seen that

H'!. A H'!, = H'!, A * H '!,. so that K = H'!, A H'!· is self-adjoint. As a

consequence. the proof that K has the same sequence of characteristic values *" 0 as Kundergoes no change.

§ 3. Properties of the system of linear integral equations (1).

We shall discuss now the system of linear integral equations (1) with matrix-kemel 11 Kij (x. y) 11 satisfying the conditions a.. b.. c. and d .. mentioned in § 1. By VII. Theorem 4. the linear transformation

I g I = H I fl in the HILBERT space [L~m) (6W. defined by

gl (x) = ~J=I hij (x) fj (x) (i = 1 •...• n).

is bounded. self-adjoint and of positive type. while by VII. Theorem 5. the linear transformation I g I = A I fl in this space. defined by

gl (x) = ~J=I.f Alj (x. y) fj (y) dy (i = 1. ...• n). l>

is completely continuous. Moreover. by condition d .• we have

HAH=HA*H.

The transformation K=AH. given by Igl =Klfl. where

gl (x) = ~J=I J Klj (x. y) fj (y) dg (i= 1 •.•.• n).

is therefore completely continuous. and. by Lemma 2. symmetrisable with respect to the transformation H. Supposing that at least one of the functions hij (x) *" 0 on a set of positive measure. so that H*"O (cf. VII. Theorem 4). all theorems in I. and in particular (by Lemma 2)

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those in I. § 5, may therefore be applied to the equation KI fl-).I fI = 1 g I in [L 2]n or, which is equivalent, to the system of equations (1) in L 2 •

Before stating results, we mention a theorem about Hand its "positive square root" Hl' •.

Theorem 1. The uniquely determined, bounded, self~adjoint trans~ formation 1 g I = Hl', 1 fl of positive type in [L2]n is determined by

(i = 1, ... , n),

where the functions hW(x) are those determined in Lemma 1. Proof. Follows immediate1y from Lemma 1 and the fact that Hl" is

uniquely determined. For the system (1) the following theorems hold:

Theorem 2. The characteristic va lues of (1) are real, and characteristic functionsets, belonging to different characteristic values, are H~orthogonal,

that is. if lP~ (x) belongs to the characteristic value )'ko lP} (x) to ).1, and ).k *' ).1, then

2~j=' J hij (x) lP} (x) lP" (x) dx = O. ö

Proof. Follows from I, Theorem 5.

Theorem 3. I{)' *' 0 is a characteristic value of (1), this system of equations has, {or a given 1 g I€ [L2]n, a solution 1 fl € [L 2]n {or those and only those 1 g I € [L2]n that are H~orthogonal to all characteristic (unctionsets belonging to the characteristic value ).. If ). *' 0 is no characteristic value of (1). the system (1) has a uniquely determined solution 1 fI € [L 2]n (or every I g I € [L2]n.

Proof. Follows from I. Theorem 12 and I, Theorem 3.

Theorem 4. I{

..Er="f Kl~) (x, x) dx *' 0, ö

where

Kl~ (x, y) = ..El=1 J Kil (x, z) Klj (z, y) dz, ö

the system (1) has a characteristic value *' O. Proof. Since, by I, Theorem 18, K has the same characteristic values

*' 0 as the se1f~adjoint transformation K = Hl'tA Hl'. with Hermitian

matrix~kernelll Kij(x.y) 11. wh ere Kij(X' y) = ~f.I=1 h~~,)(x) Aki (x, y) h~~,)(y), our system (1) has a characteristic value *' 0 if only

..Ei~j=1 JIKi j (x, y) 12 dxdy*,0 • (3) öXö

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(cf. VII, Theorem 9). An easy computation gives

I~/=I I Kil (x, y) 12 = 2f,/=1 Kil (x, y) Kit (y, x):

the inequality (3) is therefore equivalent with

2r=1 J KW (x, x) dx"=/- O. L:,.

Let now Àk ( I ÀI 1 ;::: 1 À21 ;::: ••• ) be the sequence of all characteristic values "=/- 0, each of them occurring as many times as denoted by its multiplicity, and V'i (x) a corresponding H~orthonormal sequence of

characteristic functionsets. These functionsets satisfy therefore the relations

2F',j=1 .f hij (x) V'i (x) V't (x) dx = 1, L:,.

If,/= I J hij (x) V'i (x) V'! (x) dx = 0 for k "=/-1, L:,.

. . (4)

Theorem 5 (Expansion Theorem). If

ak=(lfI, IXkl}=Ir=l! fi(X) Xi(x)dx, 1\

then

2j=1 J Kij (x, y) fl (g) dg C/l Ik Àk ak V'i (x) + pi (x) (i=1. ... ,n) (5) L:,.

for any I fl € [L2]n. where I p I = I pI (x), ... , pn (x) I satisfies Hip I = 101. that is,

(i = 1. ... , nl.

Furthermore, if P = H K has the matrix~kernel 11 Pij (x, y) 11,

If,j=1 J Pij (x, y) [i (x) fl (y) dx dy = Ik).k lak 12

• • • • (6) L:,.XL:,.

Proof. Follows from I, Theorem 15 and J, Theorem 9. Corollary. If ak=(lfI, IXkl)=O for k=1,2, ... , then HKlfI=IOI

and (H I fl. KI g I) = 0 for every I g I €[L2]n. Proof. Sin ce ak = 0 for k = 1, 2, ... , we have K I ti = I p I with

Hip I = I 0 I, hence H KI fI = 101. This implies

(H lf}, Klg!) = (It}, H K Igl) = (HKlfl, Igl) =0 for every I g I € [L2]n.

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Theorem 6. Let li [ (I = 1. 2 •... ) be the subseqttence of all positive

characteristic values. where lk,;::: 1kt ;::: . . .• and let P = H K have the matrix~kernel 11 P'J (x. y) 11. Then

lk[ = max E~J=I J PIJ (x. y) [i (x) fJ (y) dx dyl 6.x6.

~~J=I J hiJ (x) f' (x) fJ (x) dx 6.

for all I fl € [L2]n satisfying the conditions E~J=t f hiJ (x) [i (x) fJ(x) dx=j=.O 6.

and (ltI.lxk,I)= ... =(lfl.lxk,_tl)=O. For Ifl= IXk[1 themaximum is attained.

A similar statement holds for the subsequence of all negative characteristic va lues.

Proof. Follows from I. Theorem 10.

Theorem 7. Let the functionsets p~ (x) • ...• P~_I (x) (i = 1. ...• n;

all p~ (x) € L~m) (.6.» be arbitrarily given. and let

!'I = upper bound .2f.J=1 J PIJ (x. y) [i (x) f1 (y) dxdy/ 6.X6.

E7.J=J hij (x) [i (x) fJ (x) dx 6.

for all Itl € [L1]n satisfying ~~J=I f hiJ(x)[i(x)fJ(x)dx=j=. 0 and 6.

(H I fl. I PI I) = ... = (H I fl. I PI-tl) = O. The number !'I depends on I PI I.· ... 1 PI-I I. Letting now these elements run through the whole space [L 2]n. we have

1k[ = min !'I. A simtlar statement holds for the negative characteristic values. Proof. Follows from I. Theorem 11.

Theorem 8. Let 1 =j=. O. and let I 9 I € [L2]n be H~orthogonal to all characteristic functionsets of (1). belonging to the characteristic value 1 (If 1 is no characteristic value. I 9 I is therefore arbitrary in [L2]n). Then every solution I fl of (1) satisfies a relation of the form

f i ( ) gl (x) + 'Ç" 1k I ( ) + I ( ) X(/)- -.t- ~k 1 (lk- 1) ak '11k x q X (i= 1 •.•.• nl.

where ak =~=I f gi(x) Xk(x) dx for 1k =j=.1. H Iql = 101. and where ~k 6.

denotes that for those va lues of k for which lk = 1 the coeflicient of

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tp~(x) has the value ~r=1 f fi (x) x~ (x) dx. For every set of arbitrarily 1:;,

prescribed values of the latter coeflicients there exists a solution of (1). Pro of. Follows from I, Theorem 17.

Theorem 9. We have

~r=1 J KW (x, x) dx = ~k ÀL (7) 1:;,

Proof. In the proof of Theorem 4 we have already seen that

so that, since by VII, Theorem 13, the expression on the right is equal

to ~k Àt formula (7) follows.

Theorem 10 (Expansion Theorem for the elements of the matrix~ kemel). We have

(i,j= 1, •.. , n), . (8)

where Pij (x, y) € L~m) (L) (i, j = 1, ... , n) satisfies

~;=I hiq (x) pqj (x, y) = 0.. . . . . . . (9)

Pro of. We ob serve first that, by I, Theorem 18, K = H '/, A H '/· has the same sequence Àk of characteristic values *- 0 as K = AH, and that IlJfk 1= H '/, I tpk I is a corresponding orthonormal sequence of characteristic elements. Hence, by VII, Theorem 13 (1 J),

Kiq (x, y) (/) ~k Àk 1Jf1 (x) IJff (y)

with 1Jf1(x) = ~J=I h~J)(x) tp{(x), 50 that

(i, q = 1, ... , n)

~~=I h~'~) (x) Krj (x, y) = l'~,S=1 h~'~) (x) Ars (x, y) hsl (y) =

or

Z=I h~'~) (x) Kr} (x, y) (/) ~k Àk 1Jf1 (x) x{ (y). . . . (la)

Let us consider now the matrix~kernel 11 Dij (x, y) 11, belonging to the transformation D=A H'/·. Then, for every i(i= 1, ... , n) and for al most

every x € L, I di I = I d} (y), ... , d7 (y) I, where d{ (y) = Dij (x, y), belongs to [L 2]n. Hence, by BESSEL'S inequality (the system of functionsets

1Jf1 (x) is orthonormal),

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or. since

(ld;l.1 'Pkl) = .El=IJ dl (y) 'PI (y) dy = .0=IJ Dij (x. y) 'PI (y) dy = ~ ~

2:1= 1 J .E;. r= 1 Aiq (x. y) h~1) (y) hj''j (y) 'Pr (y) dy = ~

.E~=IJ Kir (x. y) 'Pr (y) dy = Äk 'Pi (x). ~

2:kÄ7c1'Pi(xW~2:1=IJIDij(x.y)12dy ... • (11) ~

for al most every x € L. Then we observe that in the HILBERT space [L~m) (LW' the elements

(i.j = 1. .... n)

are orthogonal on account of the orthogonality of the system I 'Pk I in

[L~m)(LW. Furthermore. by (11).

2:k 111 {kl W = .Ek Ä7c (.2f.j=IJI 'PI (x) 12 ·1 'PI (y) 12 dxdy) = ~x~

.Ek Ä7c (.EF=IJ I'Pi (x) 12 dx) ~ .E~j=IJIDij (x. y)12 dxdy < 00.

~ ~x~

whïch shows. since

112:%=p I {kl W = .E%=pl l l{kl W by the orthogonality. that 2:k I (k I con verg es in [L~m) (L)]n'. This implies

that. for i. j = 1 •...• n. the series .Ek Äk 'Pi (x) 'Pf (y) converges in mean. Denoting the sumfunction by 6j (x. y). we have therefore

{ij (x. y) (/) 2:k Äk 'Pi (x) IJIf (y). 50 that. writing

.E;=I 6q (x. y) h~1) (y) = Klj (x. y)-Pij (x. y). we have also

(i.j = 1. .... nl. (12)

The only thing th at remains to be proved is .E;= 1 hiq (x) pqj (x. y) = O. From (12) we deduce

.E~=I h}?) (x) Krj (x. y)-.E~=I h~l?) (x) prj (x. y) (/).Ek Äk 'Pt (x) x{ (y).

hence. comparing this with (10).

2:~=1 h}?)(X)Prj(x. y) =0 or .E~=I hir(x)Prj(X.y)=O.

This completes the proof.