Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every...

Blaschke products, Poncelet’s theorem, andOperator Theory

Pamela Gorkin

Bucknell University

October, 2015

Three questions

Numerical range

A an n ⇥ n matrix.

The numerical range of A is W (A) = {hAx , xi : kxk = 1}.

Why the numerical range?

Contains eigenvalues of A : hAx , xi = h�x , xi = �hx , xi = �.

Compare the zero matrix and the n ⇥ n Jordan block: (Here’s the2⇥ 2)

A1 =

0 00 0

�,A2 =

0 10 0

�.

W (A1) = {0},W (A2) = {z : |z | 1/2}.

Q1. What’s the numerical range of an n ⇥ n matrix A?

Starter question: What’s the numerical range of a 2⇥ 2 matrix A?

Blaschke products

B(z) = �nY

j=1

z � aj1� ajz

, where aj 2 D, |�| = 1.

Basic fact: A Blaschke product of degree n maps the unit circleonto itself n times.

Question 2: What happens when we connect the points Bidentifies on the unit circle; i.e., the n points for which B(z) = �for each � 2 T?

Starter question: If B is degree three, we connect the 3 points forwhich B(z) = � for each � 2 T; so we have triangles associatedwith points in T. Is there a connection between all of thesetriangles?

Poncelet’s theorem

Let E1 and E2 be ellipses with E1 entirely contained in E2.”Shoot” as indicated in the picture:

Maybe the ball just keeps moving, forever – never returning to thestarting point. Maybe, though, it does return to the initial point.

Poncelet’s theorem says that if you shoot according to this ruleand the path closes in n steps, then no matter where you begin thepath will close in n steps.

Question 3. Why does this happen?

Starter question. Why does this happen for triangles?

Amazing fact

These three questions are all the same.

Numerical Range

W (A) = {hAx , xi : kxk = 1}.Theorem (Elliptical range theorem)

Let A be a 2⇥ 2 matrix with eigenvalues a and b. Then thenumerical range of A is an elliptical disk with foci at a and b andminor axis given by ( tr(A?A)� |a|2 � |b|2)1/2.

Interesting consequence

Theorem (The Toeplitz-Hausdor↵ Theorem; 1918)

The numerical range of an n ⇥ n matrix is convex.

Remark: Every unitary matrix is unitarily equivalent to a diagonalmatrix, with its eigenvalues on the diagonal. If

A =

2

4�1 0 00 �2 00 0 �3

3

5

then hA1x , xi =P3

j=1 �j |xj |2, which is the convex hull of theeigenvalues.

Fact: The numerical range of a unitary matrix is the convex hullof its eigenvalues.

Blaschke products

Example. The Blaschke product with zeros at 0, 0.5 + 0.5i and�0.5 + 0.5i .

Figure: One, two and many triangles

Blaschke products

Example. The Blaschke product with zeros at 0, 0.5 + 0.5i and�0.5 + 0.5i .

Figure: A Blaschke 3-ellipse

A Blaschke 3-ellipse is an example of a Poncelet 3-ellipse.

For every a, b 2 D, there is a Poncelet 3-ellipse with those as foci.

What would these be “the same”?

Theorem (Daepp, G., Mortini)

Let B be a Blaschke product with zeros 0, a1 and a2. For � 2 T,let z1, z2 and z3 be the distinct solutions to B(z) = �. Then thelines joining zj and zk , for j 6= k , are tangent to the ellipse given by

|w � a1|+ |w � a2| = |1� a1a2|.

Conversely, every point on the ellipse is the point of tangency of aline segment that intersects T at points for which B(z1) = B(z2).

Theorem (Elliptical range theorem)

Let A be a 2⇥ 2 matrix with eigenvalues a and b. Then thenumerical range of A is an elliptical disk with foci at a and b andminor axis given by (tr(A?A)� |a|2 � |b|2)1/2.

The connection

A special class of matrices

Let a and b be the zeros of the Blaschke product.

A =

a

p1� |a|2p1� |b|2

0 b

�.

The numerical range of A is the elliptical disk with foci a and band the length of the major axis is |1� ab|.

What’s the connection? The boundary of the numerical range ofthis A is our Blaschke ellipse.

A special class of matrices

Let a and b be the zeros of the Blaschke product.

A =

a

p1� |a|2p1� |b|2

0 b

�.

The numerical range of A is the elliptical disk with foci a and band the length of the major axis is |1� ab|.What’s the connection? The boundary of the numerical range ofthis A is our Blaschke ellipse.

The takeaway

A =

a

p1� |a|2p1� |b|2

0 b

�.

1 A has a unitary 1-dilation.

2 A is a contraction; (Ax = U�

x

0

�)

3 The defect indices rank(I � A?A) = rank(I � AA?) = 1.

Look at unitary dilations of A. For � 2 T

U� =

2

4a

p1� |a|2p1� |b|2 �bp1� |a|2

0 bp

1� |b|2�p1� |a|2 ��ap1� |b|2 �ab

3

5

The takeaway

A =

a

p1� |a|2p1� |b|2

0 b

�.

1 A has a unitary 1-dilation.

2 A is a contraction; (Ax = U�

x

0

�)

3 The defect indices rank(I � A?A) = rank(I � AA?) = 1.

Look at unitary dilations of A. For � 2 T

U� =

2

4a

p1� |a|2p1� |b|2 �bp1� |a|2

0 bp1� |b|2

�p1� |a|2 ��ap1� |b|2 �ab

3

5

Paul Halmos asked, essentially, “What can unitary dilations tellyou about your operator?”

Figure: me, Sheldon Axler, Don Sarason, Paul Halmos

W (A) = \↵{W (U↵) : U↵ a unitary dilation of A}and for our special matrices

W (A) = \↵{W (U↵) : U↵ a unitary 1-dilation of A}



W (A) = \↵{W (U↵) : U↵ a unitary dilation of A}

and for our special matrices




W (A) = \↵{W (U↵) : U↵ a unitary dilation of A}and for our special matrices


• The numerical range of a unitary matrix is the convex hull of itseigenvalues.

Question: What are the eigenvalues of our matrices U�? We needto compute the characteristic polynomial.

The characteristic polynomial of zI � U� is

z(z � a)(z � b)� �(1� az)(1� bz).

The eigenvalues of U� are the points on the unit circle that satisfy

z(z � a)(z � b)

(1� az)(1� bz)= �.

• The numerical range of a unitary matrix is the convex hull of itseigenvalues.

Question: What are the eigenvalues of our matrices U�? We needto compute the characteristic polynomial.

The characteristic polynomial of zI � U� is

z(z � a)(z � b)� �(1� az)(1� bz).

The eigenvalues of U� are the points on the unit circle that satisfy

B(z) =z(z � a)(z � b)

(1� az)(1� bz)= �.

And now we also see the connection to Poncelet’s theorem!

Blaschke ! Poncelet; Poncelet ! Blaschke?

Every Blaschke 3-ellipse is a Poncelet 3-ellipse. Conversely?

Take a Poncelet 3-ellipse and the Blaschke ellipse with same foci.

They can’t both be inscribed in triangles unless...

So Blaschke 3-ellipses and Poncelet 3-ellipses are the same.

Blaschke ! Poncelet; Poncelet ! Blaschke?

Every Blaschke 3-ellipse is a Poncelet 3-ellipse. Conversely?

z1

z2

z3

w2

w3

E1

E2

Take a Poncelet 3-ellipse and the Blaschke ellipse with same foci.

They can’t both be inscribed in triangles unless...

So Blaschke 3-ellipses and Poncelet 3-ellipses are the same.

But that’s a pretty weird matrix, isnt’ it?

Actually, no it’s not.

Operator theory

H2 is the Hardy space; f (z) =P1

n=0 anzn where

P1n=0 |an|2 <1.

An inner function is a bounded analytic function on D with radiallimits of modulus one almost everywhere.

S is the shift operator S : H2 ! H2 defined by [S(f )](z) = zf (z);

The adjoint is [S?(f )](z) = (f (z)� f (0))/z .

Theorem (Beurling’s theorem)

The nontrivial invariant subspaces under S are

UH2 = {Uh : h 2 H2},

where U is an inner function.

Subspaces invariant under the adjoint, S? are KU := H2 UH2.

What’s the model space?

Theorem

Let U be inner. Then KU = H2 \ U zH2.

So {f 2 H2 : f = gzU a.e. for some g 2 H2}.

Consider KB where B(z) =Qn

j=1z�aj1�aj z

.

Consider the Szego kernel: ga(z) =1

1� az.

• hf , gai = f (a) for all f 2 H2.

• So hBh, gaj i = B(aj)h(aj) = 0 for all h 2 H2.

So gaj 2 KB for j = 1, 2, . . . , n.

If aj are distinct, KB = span{gaj : j = 1, . . . , n}.

Compressions of the shift

Consider the compression of the shift: SB : KB ! KB defined by

SB(f ) = PBS(f ),

where PB is the orthogonal projection from H2 onto KB .

Applying Gram-Schmidt to the kernels we get the Takenaka basis:Let ba(z) =

z�a1�az and

{p1� |a1|21� a1z

, ba1

p1� |a2|21� a2z

, . . .k�1Y

j=1

baj

p1� |ak |21� akz

, . . .}.

What’s the matrix representation for SB with respect to this basis?

For two zeros it’s...

A =

a

p1� |a|2p1� |b|2

0 b

�.

So A is the matrix representing SB when B has two zeros a and b.

So, the numerical range of SB is an elliptical disk, because thematrix is 2⇥ 2.

• The numerical range of our 2⇥ 2 matrix A is the intersection ofthe numerical ranges of its unitary dilations.

The picture would be triangles that intersect to give you an ellipse.

• If we take the two foci and add a zero at zero, we have aBlaschke product of degree 3.

If we join the three points B sends to each point � on the circle,we’ll see triangles that intersect to give you an ellipse.

• These ellipses are precisely the Poncelet 3-ellipses.

Does all this work for larger matrices?

The n ⇥ n matrix A is2

666666664

a1p1� |a1|2

p1� |a2|2 . . . (

Qn�1k=2(�ak))

p1� |a1|2

p1� |an|2

0 a2 . . . (Qn�1

k=3(�ak))p1� |a2|2

p1� |an|2

. . . . . . . . . . . .

0 0 0 an

3

777777775

The unitary 1-dilation of A is given by

bij =

8>><

>>:

aij if 1 i , j n,

��Qj�1

k=1(�ak)�p

1� |aj |2 if i = n + 1 and 1 j n,�Qnk=i+1(�ak)

�p1� |ai |2 if j = n + 1 and 1 i n,

�Qn

k=1(�ak) if i = j = n + 1.

An interesting special case

This class of matrices is called Sn. These matrices have unitary1-dilations, no eigenvalues of modulus 1, and are contractions.

Still, they may seem special. So here are some examples you arefamiliar with:

When the Blaschke product is B(z) = zn, the matrix A is then ⇥ n Jordan block.

Theorem (Haagerup, de la Harpe 1992)

The numerical range of the n ⇥ n Jordan block is a circular disk ofradius cos(⇡/(n + 1)).

Generalizing these results

Theorem (Gau, Wu)

For any matrix A in Sn and any point � 2 T there is an(n + 1)-gon inscribed in T that circumscribes the boundary ofW (A) and has � as a vertex.

5 curve.pdf curve.pdf

Figure: Poncelet curves

Some geometric properties remain, but these are no longer ellipses.

Generalizing these results

Theorem (Gau, Wu)

For any matrix A in Sn and any point � 2 T there is an(n + 1)-gon inscribed in T that circumscribes the boundary ofW (A) and has � as a vertex.

5 curve.pdf

curve.pdf


Some geometric properties remain, but these are no longer ellipses.

Question: When is the numerical range of a matrix in Sn an elliptical disk?

Fujimura, 2013

Let’s look at 3⇥ 3 matrices.

Note: For 2⇥ 2 we looked at ellipses inscribed in triangles.

Theorem

Let E be an ellipse. TFAE:• E is inscribed in a quadrilateral inscribed in T;• For some a, b 2 D, the ellipse E is defined by the equation

|z � a|+ |z � b| = |1� ab|s

|a|2 + |b|2 � 2

|ab|2 � 1.

Let B have zeros a, b, c and B1(z) = zB(z).

Lemma (The Composition Lemma)

A quadrilateral inscribed in T circumscribes an ellipse E i↵ E isassociated with B1 and B1 is the composition of two degree-2normalized Blaschke products.

A (very) brief look at the projective geometry side of this

Write A = H + iK with H,K Hermitian and let

LA(u, v ,w) = det(uH + vK + wI ),

where u, v ,w are viewed as line coordinates.

LA(u, v ,w) = 0 defines an algebraic curve of class n (LA hasdegree n).

The real part of this curve is called the Kippenhahn curve, C (A),and W (A) is the convex hull of C (A).

Kippenhahn gave a classification scheme depending on how LAfactors when n = 3.

Kippenhahn, 1951

LA factors into three linear factors: Then C (A) consists of threepoints.

LA factors into a linear factor and a quadratic factor: Then C (A)consists of a point �0 and an ellipse.

Keeler, Rodman, Spitkovsky, 1997

Theorem

Let A be a 3⇥ 3 matrix with eigenvalues a, b, c of the form

A =

0

@a x y0 b z0 0 c

1

A

Then W (A) is an elliptical disk if and only if all the following hold:

1 d = |x |2 + |y |2 + |z |2 > 0;

2 The number � = (c |x |2 + b|y |2 + a|z |2 � xyz)/d coincideswith at least one of the eigenvalues a, b, c ;

3 If �j denote the eigenvalues of A for j = 1, 2, 3 and � = �3

then (|�1 � �3|+ |�2 � �3|)2 � |�1 � �2|2 d .

So one of these points is special.

What KRS means to us

Theorem (G., Wagner)

Let B be have zeroes at a, b, and c and SB the correspondingcompression of the shift. TFAE:

(1)The numerical range of SB is an elliptical disk;

(2) B1(z) = zB(z) is a composition;

(3) The intersection of the closed regions bounded by thequadrilaterals connecting the points identified by B1(z) = zB(z) isan elliptical disk.

The equation of the ellipse that we obtain is (d = |x |2+ |y |2+ |z |2)

|z � b|+ |z � c | = pd + |b � c |2

versus |1� bc |q

2�|b|2�|c|21�|bc|2 .

Hmmmm...these look di↵erent! But they can’t be!








|z � b|+ |z � c | = pd + |b � c |2 versus |1� bc |

q2�|b|2�|c|21�|bc|2 .










q2�|b|2�|c|21�|bc|2 .

Hmmmm...these look di↵erent!

But they can’t be!









q2�|b|2�|c|21�|bc|2 .


Theorem

Let n 2 Z+ with n � 3. For b, c 2 D, there is at most onePoncelet n-ellipse with foci at b and c .

Let 'a(z) =a� z

1� az.

Corollary

An ellipse is a Poncelet 4-ellipse only if there exists a 2 D suchthat the foci, b and c , satisfy b = 'a(c). Conversely, givenb, c 2 D with b = 'a(c) for some a 2 D, there exists a uniquePoncelet 4-ellipse with foci at b and c .

If the corresponding Blaschke product has zeros 0, a, b and c , then'a permutes the zeros and there is very little choice for a.

Familiar consequences?

Theorem (Brianchon’s theorem)

If a hexagon circumscribes an ellipse, then the diagonals of thehexagon meet in one point.


An ellipse is a Poncelet 4-ellipse if and only if there exists a pointa 2 D such that the diagonals of every circumscribing quadrilateralpass through a.

A Natural Question: What happens for degree 6?

Example 1. Let B1 = C1 � D1, where

C1(z) = z

✓z � a

1� az

◆2

and D1(z) = z2.

If B(z) = B1(z)/z , then W (AB) is an elliptical disk.

B1: An ellipse B2: Not an ellipse.


Example 2. Let B2 = C2 � D2, where

C2(z) = z2 and D2(z) = z

✓z � (0.5 + 0.5 i)

1� (0.5� 0.5 i)z

◆2

.

If B(z) = B2(z)/z , then W (AB) is not an elliptical disk.

Infinite Blaschke products

Theorem (Chalendar, G., Partington)

Let I be an inner function. Then the closure of the numericalrange of SI satisfies

W (SI ) =\

↵2TW (U I

↵),

where the U I↵ are the unitary 1-dilations of SI (or, equivalently, the

rank-1 Clark perturbations of SzI ).

For some functions, we get an infinite version of Poncelet’stheorem.

Vector-valued version of this theorem (Benhida, G., and Timotin)

Further generalizations

Let DT = (1� T ?T )1/2 (the defect operator) and DT = DTH(the defect space).

What if the dimension of DT = DT? = n > 1?

Bercovici and Timotin showed that

W (T ) =T{W (U) : U a unitary n dilation of T} and generalized

some of the geometric properties to the infinite dimensional case(for example, W (T ) has no corners).

Directions for further research

1 Study the numerical range of other operators;

2 When is 0 in the numerical range?

3 Michel Crouzeix’s 2006 paper, “Open problems on thenumerical range and functional calculus” contains severalquestions that remain open.

Conjecture: For any polynomial p the inequality holds:

kp(A)k 2max |p(z)|z2W (A)

(The constant 11.08 works.)

Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every...

Documents

Transcript of Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every...