The numerical range, Blaschke products and … · AIM August 2017. Two results about ellipses....

The numerical range, Blaschke products andCompressions of the shift operator

Pamela Gorkin

Bucknell University

AIM August 2017

Two results about ellipses

Blaschke products and Blaschke ellipses

B(z) = λ

n∏j=1

z − aj1− ajz

, where aj ∈ D, |λ| = 1.

Basic fact: A Blaschke product of degree n maps the unit circleonto itself n times; the argument is increasing and B(z) = λ hasexactly n distinct solutions for each λ ∈ T.

Question: What happens when we connect the points B identifieson the unit circle; i.e., the n points for which B(z) = λ for eachλ ∈ T?

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

Blaschke products

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

One, two and many triangles

Blaschke products

A Blaschke 3-ellipse

First theorem

Theorem (Daepp, G., Mortini)

Let B be a Blaschke product with zeros 0, a1 and a2. For λ ∈ T,let z1, z2 and z3 be the distinct solutions to B(z) = λ. 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).

The Blaschke curve

Definition

Let B be a Blaschke product with zeros 0 = a1, a2, . . . , an+1. Forλ ∈ T, let Pλ be the closed convex hull of the n + 1 distinctsolutions of B(z) = λ. Then the boundary of ∩λPλ is called theBlaschke curve associated with B.

Remarks. We assume B(0) = 0, but that is not necessary. Ifϕa(z) = z−a

1−az , then

B1 = ϕB(0) ◦ B

identifies the same set of points as B(0).

Poncelet’s theorem, 1813

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.

New proof Halbeisen and Hungerbuhler, 2015!

Theorem (Pascal, 1639-40; Braikenridge-Maclaurin)

If a hexagon is inscribed in a nondegenerate conic, the intersectionpoints of the three pairs of opposite sides are collinear and distinct.

Conversely, if at least five vertices of a hexagon are in generalposition and the hexagon has the property that the points ofintersection of the three pairs of opposite sides are collinear, thenthe hexagon is inscribed in a unique nondegenerate conic.

Illustration of Pascal’s theorem

The dual of Pascal’s theorem

Theorem (Brianchon, 167 years later!)

If a hexagon circumscribes a nondegenerate conic, then the threediagonals are concurrent and distinct.

Conversely, if at least five of the sides of a hexagon are in generalposition and the hexagon has the property that its three diagonalsare concurrent, then the six sides are tangent to a uniquenondegenerate conic.

Illustration of Brianchon’s theorem

Question. What is the connection?

Starter question. Why does this happen for triangles?

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

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

What’s the connection to the numerical range?

|w − a1|+ |w − a2| = |1− a1a2|.

W (A) = {〈Ax , x〉 : ‖x‖ = 1}.

Theorem (Elliptical range theorem)

Let A be a 2× 2 matrix with eigenvalues a and b. Then W (A) isan elliptical disk with foci at a and b and minor axis given by(tr(A?A)− |a|2 − |b|2)1/2.

What’s the connection to the numerical range?

|w − a1|+ |w − a2| = |1− a1a2|.

W (A) = {〈Ax , x〉 : ‖x‖ = 1}.

Theorem (Elliptical range theorem)

Let A be a 2× 2 matrix with eigenvalues a and b. Then W (A) isan elliptical disk with foci at a and b and minor 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.

1− |a|2√

1− |b|20 b

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

The connection: The boundary of the numerical range of this A isthe Blaschke ellipse associated with B(z) = zB(z), where B haszeros at a and b.

But these are very special ellipses: all are Poncelet ellipses.

A special class of matrices

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

1− |a|2√

1− |b|20 b

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

The connection: The boundary of the numerical range of this A isthe Blaschke ellipse associated with B(z) = zB(z), where B haszeros at a and b.

But these are very special ellipses: all are Poncelet ellipses.

The takeaway

1− |a|2√

1− |b|20 b

Let S be an operator on H and let K ⊃ H be Hilbert spaces. ThenT is a dilation of S if PHT |H = S (S is a compression of T ).

Halmos: Every contraction T has a unitary dilation; letDT = (I − T ?T )1/2

[T DT?

DT −T ?

1 A is a contraction with all eigenvalues in D;2 rank(I − A?A) = rank(I − AA?) = 1.3 A has a unitary 1-dilation; (A is in upper-left corner of a 3× 3

unitary matrix).

These properties hold in a class of operators that we nowinvestigate.

Question 1. Is Crouzeix’s conjecture true for this class ofoperators?

Special properties of A that might be useful.

1− |a|2√

1− |b|20 b

Look at unitary dilations of A. For λ ∈ T

1− |a|2√

1− |b|2 −b√

1− |a|20 b

√1− |b|2

1− |a|2 −λa√

1− |b|2 λab

Special properties of A that might be useful.

1− |a|2√

1− |b|20 b

Look at unitary dilations of A. For λ ∈ T

1− |a|2√

1− |b|2 −b√

1− |a|20 b

√1− |b|2

1− |a|2 −λa√

1− |b|2 λab

Paul Halmos and his conjecture

me, S. Axler, D. Sarason (1933 – 2017), P. Halmos (1916 – 2006)

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

W (T ) =⋂{W (U) : U a unitary dilation of T}?

and for our special matrices

W (A) =⋂α∈T{W (Uα) : Uα a unitary 1-dilation of A}?

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

What are the eigenvalues of our matrices Uλ? We need tocompute the characteristic polynomial.

The characteristic polynomial of zI − Uλ is

z(z − a)(z − b)− λ(1− az)(1− bz).

The eigenvalues of Uλ are the points on the unit circle that satisfy

z(z − a)(z − b)

(1− az)(1− bz)= λ.

What are the eigenvalues of our matrices Uλ? We need tocompute the characteristic polynomial.

z(z − a)(z − b)− λ(1− az)(1− bz).

z(z − a)(z − b)

(1− az)(1− bz)= λ.

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

z(z − a)(z − b)− λ(1− az)(1− bz).

B(z) =z(z − a)(z − b)

(1− az)(1− bz)= λ.

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.

Then look at the picture!

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

How this works for degree-n Blaschke products

Operator theory

H2 is the Hardy space; f (z) =∑∞

n=0 anzn where

∑∞n=0 |an|2 <∞.

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 ∈ H2},

where U is a (nonconstant) 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 ∈ H2 : f = Ugz a.e. for some g ∈ H2}.

Finite-dimensional model spaces: KB where B(z) =∏n

j=1z−aj1−ajz .

Consider the Szego kernel: ga(z) =1

1− az.

• 〈f , ga〉 = f (a) for all f ∈ H2.

• So 〈Bh, gaj 〉 = B(aj)h(aj) = 0 for all h ∈ H2.

So gaj ∈ KB for j = 1, 2, . . . , n.

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

Theorem

j=1z−aj1−ajz .

1− az.

• 〈f , ga〉 = f (a) for all f ∈ H2.

So gaj ∈ KB for j = 1, 2, . . . , n.

Theorem

j=1z−aj1−ajz .

1− az.

• 〈f , ga〉 = f (a) for all f ∈ H2.

So gaj ∈ KB for j = 1, 2, . . . , n.

Theorem

j=1z−aj1−ajz .

1− az.

• 〈f , ga〉 = f (a) for all f ∈ H2.

So gaj ∈ KB for j = 1, 2, . . . , n.

Compressions of the shift (Sarason, ’67; Sz.-Nagy, Foias)

Compression of the shift: SB : KB → KB defined by

SB(f ) = PB(S(f )),

where PB is the orthogonal projection from H2 onto KB .

Let Sm denote compressions of the shift (or model spaceoperators) on a space of dimension m.

Theorem (Nakazi, Takahashi, 1995)

Let A be an n× n cnu contraction with all eigenvalues in D. Let mbe the degree of the minimal polynomial of A andd = rank(I − A?A). Then A can be extended to a matrix of theform B ⊕ · · · ⊕ B (d times) where B is in Sm and the minimalpolynomial of B is the same as that of A.

Finding a matrix representation

We need an orthonormal basis: a1, . . . , an zeros of B (assumedistinct).

Apply Gram-Schmidt to the kernels to get theTakenaka-Malmquist basis: Let ϕa(z) = z−a

1−az and

1− |a1|21− a1z

, ϕa1

√1− |a2|2

1− a2z, . . .

k−1∏j=1

√1− |ak |2

1− akz, . . .}.

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

For two zeros it’s...

1− |a|2√

1− |b|20 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.

And the boundary of the numerical range is the Blaschke ellipspeassociated with B(z) = zB(z).

What about the n × n case?

1− |a|2√

1− |b|20 b

1− |a|2√

1− |b|20 b

The n × n matrix A is

1− |a1|2√

1− |a2|2 . . . (∏n−1

k=2(−ak))√

1− |a1|2√

1− |an|2

0 a2 . . . (∏n−1

k=3(−ak))√

1− |a2|2√

1− |an|2

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

0 0 0 an

For each λ ∈ T, we get a unitary 1-dilation of A:

aij if 1 ≤ i , j ≤ n,

λ(∏j−1

k=1(−ak))√

1− |aj |2 if i = n + 1 and 1 ≤ j ≤ n,(∏nk=i+1(−ak)

)√1− |ai |2 if j = n + 1 and 1 ≤ i ≤ n,

λ∏n

k=1(−ak) if i = j = n + 1.

Everything that was true about Uλ before is still true

[A stuff(λ)

stuff(λ) stuff(λ)

1 The eigenvalues of Uλ are the values B(z) := zB(z) maps to λ;

2 W (Uλ) is the closed convex hull of the points zB(z) identifies.

3 W (A) ⊆⋂{W (Uλ) : λ ∈ D}.

If V = [In, 0] be n × (n + 1), then V tx =

], ‖V tx‖ = 1,A = VUλV

〈Ax , x〉 = 〈VUλVtx , x〉 = 〈UλV

tx ,V tx〉

∈W (Uλ).

One half of Halmos’s 1964 conjecture is easy!

[A stuff(λ)

stuff(λ) stuff(λ)

1 The eigenvalues of Uλ are the values B(z) := zB(z) maps to λ;

2 W (Uλ) is the closed convex hull of the points zB(z) identifies.

3 W (A) ⊆⋂{W (Uλ) : λ ∈ D}.

If V = [In, 0] be n × (n + 1), then V tx =

], ‖V tx‖ = 1,A = VUλV

〈Ax , x〉 = 〈VUλVtx , x〉 = 〈UλV

tx ,V tx〉 ∈W (Uλ).

One half of Halmos’s 1964 conjecture is easy!

Our operators

The class Sn are compressions of the shift operator to ann-dimensional space:

These matrices have no eigenvalues of modulus 1, and arecontractions (completely non-unitary contractions) withrank(I − T ?T ) = 1 = rank(I − TT ?) (with unitary 1-dilations).

Recall: B a finite Blaschke product,

KB = H2 BH2 = H2 ∩ BzH2.

SB(f ) = PB(S(f )) where f ∈ KB ,PB : H2 → KB .

PB(g) = BP−(Bg) = B(I − P+)(Bg),

P− the orthogonal projection for L2 onto L2 H2 = zH2.

Special properties of W (SB): The Poncelet property

Theorem (Gau, Wu)

For T ∈ Sn and any point λ ∈ T there is an (n + 1)-gon inscribedin T that circumscribes the boundary of W (T ) and has λ as avertex.

5 curve.pdf

curve.pdf

Poncelet curves

These are Poncelet curves. Some geometric properties remain, butthese are not necessarily ellipses. To better understand theseoperators, we look at the proof.

Special properties of W (SB): The Poncelet property

Theorem (Gau, Wu)

For T ∈ Sn and any point λ ∈ T there is an (n + 1)-gon inscribedin T that circumscribes the boundary of W (T ) and has λ as avertex.

5 curve.pdf

curve.pdf

Poncelet curves

These are Poncelet curves. Some geometric properties remain, butthese are not necessarily ellipses. To better understand theseoperators, we look at the proof.

An example before the proof.

When the Blaschke product is B(z) = zn, the matrix representingSB is the n × n Jordan block.

Theorem (Haagerup, de la Harpe 1992)

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

The boundary of these numerical ranges are all Poncelet circles.

Proof of the Haagerup-de la Harpe’s computation

Use Gau and Wu’s result:

The numerical range of Szn is the intersection of the numericalranges of its unitary 1-dilations.

The numerical range of a unitary 1-dilation is the convex hull ofthe points identified by B(z) = zB(z) = z · zn = zn+1.

The points are solutions to zn+1 = λ, as λ ranges over points in T.

Then use zn+1 = 1 to find the radius of the circle that is theboundary of the numerical range.

Proof of W (SB) = ∩λ∈TW (Uλ)

Step 1. Show the eigenvalues, wj , of Uλ are distinct.

Step 2. Show A =∑n+1

j=1 wjV?1 EjjV1 and

∑n+1j=1 V ?

1 EjjV1 = In;

Step 3. Find x ∈⋂n−1

j=1 ker(V ?1 EjjV1) with ‖x‖ = 1.

Step 4. Compute

〈Ax , x〉 = wn 〈V ?1 EnnV1x , x〉︸︷︷︸

+wn+1 〈V ?1 E(n+1)(n+1)V1x , x〉︸︷︷︸

Note s, t ≥ 0 and s + t = 1. So the line segment joining wn

and wn+1 intersects W (A).

Step 5. Show the line segment meets W (T ) at exactly one point.

j=1 wjV?1 EjjV1 and

∑n+1j=1 V ?

1 EjjV1 = In;

Step 4. Compute

〈Ax , x〉 = wn 〈V ?1 EnnV1x , x〉︸︷︷︸

+wn+1 〈V ?1 E(n+1)(n+1)V1x , x〉︸︷︷︸

j=1 wjV?1 EjjV1 and

∑n+1j=1 V ?

1 EjjV1 = In;

Step 4. Compute

〈Ax , x〉 = wn 〈V ?1 EnnV1x , x〉︸︷︷︸

+wn+1 〈V ?1 E(n+1)(n+1)V1x , x〉︸︷︷︸

j=1 wjV?1 EjjV1 and

∑n+1j=1 V ?

1 EjjV1 = In;

Step 4. Compute

〈Ax , x〉 = wn 〈V ?1 EnnV1x , x〉︸︷︷︸

+wn+1 〈V ?1 E(n+1)(n+1)V1x , x〉︸︷︷︸

j=1 wjV?1 EjjV1 and

∑n+1j=1 V ?

1 EjjV1 = In;

Step 4. Compute

〈Ax , x〉 = wn 〈V ?1 EnnV1x , x〉︸︷︷︸

+wn+1 〈V ?1 E(n+1)(n+1)V1x , x〉︸︷︷︸

Note s, t ≥ 0 and s + t = 1.

So the line segment joining wn

j=1 wjV?1 EjjV1 and

∑n+1j=1 V ?

1 EjjV1 = In;

Step 4. Compute

〈Ax , x〉 = wn 〈V ?1 EnnV1x , x〉︸︷︷︸

+wn+1 〈V ?1 E(n+1)(n+1)V1x , x〉︸︷︷︸

j=1 wjV?1 EjjV1 and

∑n+1j=1 V ?

1 EjjV1 = In;

Step 4. Compute

〈Ax , x〉 = wn 〈V ?1 EnnV1x , x〉︸︷︷︸

+wn+1 〈V ?1 E(n+1)(n+1)V1x , x〉︸︷︷︸

Application of function theory to T ∈ Sn

Theorem (General theorem, Choi and Li, 2001)

Let T be a contraction.

W (T ) =⋂{W (U) : U a unitary dilation of T on H ⊕ H}.

Theorem (Special theorem, Gau and Wu, 1995)

Let B be a finite Blaschke product.

W (SB) =⋂{W (U) : U a unitary 1-dilation of SB}.

Theorem (Two polygon theorem)

Given two (convex) (n + 1)-gons P1 and P2 inscribed in T withinterspersed vertices, there is a unique (up to unitary equivalence)operator SB with B of degree n and W (SB) circumscribed by thetwo polygons.

Find B: Map the problem over to the upper half-plane;

Points map to aj and xj on R and the points are interspersed;

Compute F (z) =n+1∏j=1

z − xjz − aj

F is strongly real; i.e., F (H±) = H±. Map back to get B.

This Blaschke product gives you all tangent lines.

(G.-Rhoades, Courtney-Sarason, Chalendar-G.-Partington-Ross;Semmler-Wegert for minimal degree).

The advantages of this approach

You can “find” the boundary of the numerical range: Let λ ∈ T.(Dropping λ:)

Fλ(z) =B(z)/z

B(z)− λ=

n∑j=1

z − zj.

The line segment joining zj and zj+1 is tangent to the boundary ofthe numerical range at the point

mj+1zj + mjzj+1

mj + mj+1.

And we can compute mj .

This has to be good for something!

Question 2: When is the numerical range of SB elliptical?

Application to decomposition: Fujimura, 2013

For 2× 2 matrices we looked at ellipses inscribed in triangles.What about quadrilaterals?

Theorem

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

|z − a|+ |z − b| = |1− ab|

√|a|2 + |b|2 − 2

|ab|2 − 1.

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

Lemma (The Composition Lemma)

A quadrilateral inscribed in T circumscribes an ellipse E iff E isassociated with B and B is the composition of two degree-2Blaschke products.

Theorem

|z − a|+ |z − b| = |1− ab|

√|a|2 + |b|2 − 2

|ab|2 − 1.

Theorem

|z − a|+ |z − b| = |1− ab|

√|a|2 + |b|2 − 2

|ab|2 − 1.

Theorem

|z − a|+ |z − b| = |1− ab|

√|a|2 + |b|2 − 2

|ab|2 − 1.

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

P2(C) pts. (x , y , z) 6= (0, 0, 0) and (x ′, y ′, z ′) are identified if thereexists λ 6= 0 with

x = λx ′, y = λy ′, z = λz ′.

C is embedded in P2(R) via x + iy 7→ (x , y , 1).

A homogeneous polynomial p defines an algebraic curve C ofdegree equal to the degree of p via p(x , y , z) = 0.

The tangent lines to C (ux + vy + wz = 0) satisfy anotherequation L(u, v ,w) = 0 (the dual or tangential equation). Thedegree of L is the class of C.

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

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

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, theKippenhahn curve, C (A), is the dual curve of LA.

W (A) is the convex hull of the real points of C (A): W (A) is theconvex hull of{a + bi : a, b ∈ R, ua + vb + w = 0 is tangent to LA(u, v ,w) = 0}.

Kippenhahn (1951) gave a classification scheme depending on howLA factors when n = 3.

Keeler, Rodman, Spitkovsky, 1997

Theorem

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

a x y0 b z0 0 c

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

3 If λj denote the eigenvalues of A for j = 1, 2, 3 and λ = λ3then (|λ1 − λ3|+ |λ2 − λ3|)2 − |λ1 − λ2|2 ≤ d .

Minor axis length =√d . One eigenvalue is special!

Keeler, Rodman, Spitkovsky, 1997

Theorem

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

a x y0 b z0 0 c

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

3 If λj denote the eigenvalues of A for j = 1, 2, 3 and λ = λ3then (|λ1 − λ3|+ |λ2 − λ3|)2 − |λ1 − λ2|2 ≤ d .

Minor axis length =√d . One eigenvalue is special!

(For tridiagonal 3× 3, see Glader, Kurula, Lindstrom, 2017.)

What KRS means to us

Theorem (G., Wagner)

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

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

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

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

Familiar consequences

Theorem (Brianchon’s theorem)

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

Theorem (G., Wagner)

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

T or F? W (SB) elliptical if and only if B is decomposable.

False! Degree 5 already shows this can’t work.

Consider Sz4 . This has circular numerical range. But B(z) = z5

and that cannot be decomposed; that is, B 6= C ◦D with C and Dboth of degree greater than one.

T or F? W (SB) elliptical if and only if B is decomposable.

False! Degree 5 already shows this can’t work.

Consider Sz4 . This has circular numerical range. But B(z) = z5

and that cannot be decomposed; that is, B 6= C ◦D with C and Dboth of degree greater than one.

What happens for degree 6?

Example 1. Let B1 = C1 ◦ D1, where

C1(z) = z

(z − a

1− az

and D1(z) = z2.

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

Poncelet curves

Example 2. Let B2 = C2 ◦ D2, where

C2(z) = z

(z − .5

1− .5z

)and D2(z) = z3.

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

Not an ellipse

Compressions of the shift and inner functions

General inner functions

An inner function is a bounded analytic function with radial limitsof modulus one a.e.

Theorem (Frostman’s theorem.)

Let I be an inner function. ThenI − a

1− aIis a Blaschke product for

a.e. a ∈ D.

Corollary

Blaschke products are uniformly dense in the set of inner functions.

For this reason, we will focus on infinite Blaschke products:

B(z) = λ

∞∏j=1

−aj|aj |

z − aj1− ajz

where∑

(1− |aj |) <∞.

For T a completely nonunitary contraction with a unitary1-dilation

1 Every eigenvalue of T is in the interior of W (T );

2 W (T ) has no corners in D.

Let θ be an inner function.

As before, Kθ = H2 θH2 and Sθ : Kθ → Kθ is defined by

Sθ(f ) = Pθ(Sf ) = θP−(θzf ).

Unitary 1-dilations on K = H ⊕ C.Let S denote the shift operator.

Let M1 = CS?(θ) = {γ θ(z)−θ(0)z : γ ∈ C} and N1 = Kθ M1.

Let M2 = C(θθ(0)− 1

)and N2 = Kθ M2.

Use the first decomposition as domain and the second as range

Sθ(γS?θ + w) = γ(θθ(0)− 1)θ(0) + Sw .

So, there exists λ ∈ D with

[λ 00 S

]and U =

λ 0 α√

1− |λ|20 S 0

1− |λ|2 0 −αβλ

.If θ(0) = 0, then λ = 0.

Clark, 1972

Let A = SB be the compression of the shift on KB . As before, theunitary dilations can be parametrized.

But there’s another way.

Define Uλ on KB by

Uλ(f ) =

{zf (z) if f ⊥ B

λ if f = B.

(Clark, Ahern) These are exactly the unitary 1-dilations of SB (andthe rank-1 perturbations of SB).

Clark, 1972

Let A = SB be the compression of the shift on KB . As before, theunitary dilations can be parametrized. But there’s another way.

Define Uλ on KB by

Uλ(f ) =

{zf (z) if f ⊥ B

λ if f = B.

(Clark, Ahern) These are exactly the unitary 1-dilations of SB (andthe rank-1 perturbations of SB).

Theorem (Gau, Wu)

If the set of singularities on T of B is countable, then for eachλ ∈ T, Uλ is unitarily equivalent to diag (dn) with (dn) satisfyingB?(dn) = λ for all n. In this case, each side of the (infinite,convex) polygon formed by the dn intersects W (A) at a singlepoint.

Connection to boundary interpolation! (Starting with G. T. Cargo,E. Decker, A. Nicolau, G.-Mortini, Sarason, Bolotnikov).

Theorem (Chalendar, G., Partington)

Let B be an infinite Blaschke product. Then the closure of thenumerical range of SB satisfies

W (SB) =⋂α∈T

W (UBα ),

where the UBα are the unitary 1-dilations of SB (or, equivalently,

the rank-1 Clark perturbations of SB).

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

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 ) =⋂{W (U) : U a unitary n − dilation of T}.

Wilhelm Blaschke

http://www.mathe.tu-freiberg.de/fakultaet/

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