The numerical range, Blaschke products and Compressions of ...

The numerical range, Blaschke products andCompressions of the shift operator

Pamela Gorkin

Bucknell University

July 2017

Three questions

Numerical range

A an n × n matrix.

The numerical range of A is W (A) = {〈Ax , x〉 : ‖x‖ = 1}.

Why the numerical range?

Contains eigenvalues of A : 〈Ax , x〉 = 〈λx , x〉 = λ〈x , x〉 = λ.

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

[0 00 0

],A2 =

[0 10 0

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

Numerical range

A an n × n matrix.

[0 00 0

],A2 =

[0 10 0

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

Numerical range

A an n × n matrix.

[0 00 0

],A2 =

[0 10 0

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

Numerical range

A an n × n matrix.

[0 00 0

],A2 =

[0 10 0

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

Q1. Which sets can be 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) = 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 2: What happens when we connect the points Bidentifies on the unit circle; i.e., the n points for which B(z) = λfor each λ ∈ 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?

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 3. What does this theorem have to do with complexanalysis?

Starter question. Why does this happen for triangles?

Amazing fact

Blaschke products connect these three questions

Numerical Range

W (A) = {〈Ax , x〉 : ‖x‖ = 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.

Consequence:

Theorem (The Toeplitz-Hausdorff Theorem; 1918)

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

Numerical Range

W (A) = {〈Ax , x〉 : ‖x‖ = 1}.

Consequence:

Theorem (The Toeplitz-Hausdorff 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

λ1 0 00 λ2 00 0 λ3

then 〈A1x , x〉 =

∑3j=1 λj |xj |2, which is the convex hull of the

eigenvalues.

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 .

One, two and many triangles

Blaschke products

A Blaschke 3-ellipse

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 would these be “the same”?

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).

What would these be “the same”?

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 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|.

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.

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|.

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

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 ).

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

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

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 ).

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

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

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

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

W (T ) = ∩α{W (Uα) : Uα a unitary dilation of T}

and for our special matrices

W (A) = ∩α{W (Uα) : Uα a unitary 1-dilation of A}

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

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

z(z − a)(z − b)− λ(1− az)(1− bz).

z(z − a)(z − b)

(1− az)(1− bz)= λ.

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.

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

Actually, no it’s not.

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

Actually, no it’s not.

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}.

Consider 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

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

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

where PB is the orthogonal projection from H2 onto KB .

Applying Gram-Schmidt to the kernels we get theTakenaka-Malmquist basis: Let ba(z) = z−a

1−az and

1− |a1|21− a1z

√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.

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 polygon formed with the points zB(z) identifies.

3 W (A) ⊆⋂{W (Uλ) : λ ∈ D}.

Let V = [In, 0] be n × (n + 1). Then V tx =

], ‖V tx‖ = 1 and

A = VUλVt .

〈Ax , x〉 = 〈VUλVtx , x〉 = 〈UλV

tx ,V tx〉.

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 (with unitary 1-dilations).

Recall: B be 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.

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

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.

All the numerical ranges have 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

Some geometric properties remain, but these are no longer ellipses.How would the proof go? Start with λ and find the other vertices.Then show that this polygon circumscribes W (T ).

All the numerical ranges have 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

Some geometric properties remain, but these are no longer ellipses.How would the proof go? Start with λ and find the other vertices.Then show that this polygon circumscribes W (T ).

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; find

x ∈⋂n−1

j=1 ker(V ?1 EjjV1).

Step 3. 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.

Step 4. Show the line segment meets W (T ) at exactly one point.

j=1 wjV?1 EjjV1 and

∑n+1j=1 V ?

1 EjjV1 = In; find

x ∈⋂n−1

Step 3. 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.

j=1 wjV?1 EjjV1 and

∑n+1j=1 V ?

1 EjjV1 = In; find

x ∈⋂n−1

Step 3. 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.

j=1 wjV?1 EjjV1 and

∑n+1j=1 V ?

1 EjjV1 = In; find

x ∈⋂n−1

Step 3. 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.

Application of function theory to T ∈ Sn

Theorem (General theorem, Choi and Li, 2001)

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

Theorem (Special theorem, Gau and Wu, 1995)

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

Know W (T ) ⊆⋂{W (Uλ) : λ ∈ T};

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

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

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 λ0 and an ellipse.

LA irreducible gives two other cases.

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 .

So one of these points 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 .

So one of these points is special.

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.

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

|z − b|+ |z − c | =√d + |b − c|2

versus |1− bc|√

2−|b|2−|c|21−|bc|2 .

|z − b|+ |z − c | =√d + |b − c|2 versus |1− bc|

√2−|b|2−|c|21−|bc|2 .

|z − b|+ |z − c | =√d + |b − c|2 versus |1− bc|

√2−|b|2−|c|21−|bc|2 .

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 ∈ 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

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

The geometry of infinite Blaschke products

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.

Infinite Blaschke products and two decompositions

Let S denote the shift operator.

Unitary 1-dilations on K = H ⊕ C.

[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.

Theorem (Clark, 1972)

If θ(0) = 0 all unitary 1-dilations of Sθ are equivalent to rank 1perturbations of Szθ.

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}.

Some final comments

1 Michel Crouzeix 2006: “Open problems on the numericalrange and functional calculus’.”

Conjecture (2004): For any polynomial p ∈ C[z ] and A ann × n matrix the inequality holds:

‖p(A)‖ ≤ 2 max |p(z)|z∈W (A)

2 Study the numerical range of other operators;

3 When is 0 in the numerical range?

4 When is the numerical range circular? elliptical?

Examples of what is known

• (de Barra, Giles, Sims) T compact, W (T ) closed iff0 ∈W (T ).

• (Bourdon, Shapiro, Higdon)T 6= I a composition op., then0 ∈W (T );

• (Crouzeix) Best constant is between 2 and 1 +√

• (Badea, Crouzeix, Delyon) If W (A) is a disk, this is known.

• (Glader, Kurula, Lindstrom) 3× 3 tridiagonal matrices,constant diagonals.

Thank you! Kiitos! Because “Yksi kieli ei ikina riita”

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

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