blaschke products, poncelet's theorem, and operator theory · 2019. 8. 16. · conversely,...
TRANSCRIPT
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Blaschke products, Poncelet’s theorem, andOperator Theory
Pamela Gorkin
Bucknell University
October, 2015
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Three questions
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Three questions
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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}.
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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}.
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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}.
![Page 7: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/7.jpg)
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}.
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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?
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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.
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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?
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Poncelet’s theorem
Let E1 and E2 be ellipses with E1 entirely contained in E2.”Shoot” as indicated in the picture:
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Maybe the ball just keeps moving, forever – never returning to thestarting point. Maybe, though, it does return to the initial point.
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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.
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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.
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Question 3. Why does this happen?
Starter question. Why does this happen for triangles?
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Amazing fact
These three questions are all the same.
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Why?
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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.
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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.
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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.
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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
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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
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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.
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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.
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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.
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The connection
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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.
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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.
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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
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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
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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}
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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}
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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}
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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}
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• 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)= �.
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• 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)= �.
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• 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)= �.
![Page 52: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/52.jpg)
Figure: One, two and many triangles
![Page 53: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/53.jpg)
And now we also see the connection to Poncelet’s theorem!
![Page 54: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/54.jpg)
Figure: One, two and many triangles
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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.
![Page 56: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/56.jpg)
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.
![Page 57: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/57.jpg)
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.
![Page 58: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/58.jpg)
But that’s a pretty weird matrix, isnt’ it?
Actually, no it’s not.
![Page 59: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/59.jpg)
But that’s a pretty weird matrix, isnt’ it?
Actually, no it’s not.
![Page 60: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/60.jpg)
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.
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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}.
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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}.
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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}.
![Page 64: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/64.jpg)
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}.
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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?
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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.
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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.
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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.
![Page 69: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/69.jpg)
• 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?
![Page 70: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/70.jpg)
• 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?
![Page 71: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/71.jpg)
• 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?
![Page 72: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/72.jpg)
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.
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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)).
![Page 74: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/74.jpg)
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)).
![Page 75: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/75.jpg)
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)).
![Page 76: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/76.jpg)
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.
![Page 77: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/77.jpg)
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.
![Page 78: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/78.jpg)
Question: When is the numerical range of a matrix in Sn an elliptical disk?
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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.
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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.
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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.
![Page 82: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/82.jpg)
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.
![Page 83: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/83.jpg)
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.
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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.
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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.
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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!
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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 |
q2�|b|2�|c|21�|bc|2 .
Hmmmm...these look di↵erent! But they can’t be!
![Page 88: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/88.jpg)
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 |
q2�|b|2�|c|21�|bc|2 .
Hmmmm...these look di↵erent!
But they can’t be!
![Page 89: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/89.jpg)
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 |
q2�|b|2�|c|21�|bc|2 .
Hmmmm...these look di↵erent! But they can’t be!
![Page 90: Blaschke products, Poncelet's theorem, and Operator Theory · 2019. 8. 16. · Conversely, every point on the ellipse is the point of tangency of a line segment that intersects T](https://reader035.vdocument.in/reader035/viewer/2022081614/5fc029e1256f39280c18d60e/html5/thumbnails/90.jpg)
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.
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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 2 D such that the diagonals of every circumscribing quadrilateralpass through a.
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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.
Figure: Poncelet curves
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.
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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)
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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).
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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.)