The numerical range, Blaschke products and … · AIM August 2017. Two results about ellipses....
Transcript of The numerical range, Blaschke products and … · AIM August 2017. Two results about ellipses....
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The numerical range, Blaschke products andCompressions of the shift operator
Pamela Gorkin
Bucknell University
AIM August 2017
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Two results about ellipses
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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.
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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?
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Blaschke products
Example. The Blaschke product with zeros at 0, 0.5 + 0.5i and−0.5 + 0.5i .
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 .
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 .
A Blaschke 3-ellipse
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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).
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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).
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Poncelet’s theorem, 1813
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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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
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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
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Question. What is the connection?
Starter question. Why does this happen for triangles?
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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.
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What’s the connection to the numerical range?
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).
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.
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What’s the connection to the numerical range?
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).
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.
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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√
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.
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A special class of matrices
Let a and b be the zeros of the Blaschke product.
A =
[a√
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.
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The takeaway
A =
[a√
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
UT =
[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).
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These properties hold in a class of operators that we nowinvestigate.
Question 1. Is Crouzeix’s conjecture true for this class ofoperators?
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Special properties of A that might be useful.
A =
[a√
1− |a|2√
1− |b|20 b
].
Look at unitary dilations of A. For λ ∈ T
Uλ =
a√
1− |a|2√
1− |b|2 −b√
1− |a|20 b
√1− |b|2
λ√
1− |a|2 −λa√
1− |b|2 λab
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Special properties of A that might be useful.
A =
[a√
1− |a|2√
1− |b|20 b
].
Look at unitary dilations of A. For λ ∈ T
Uλ =
a√
1− |a|2√
1− |b|2 −b√
1− |a|20 b
√1− |b|2
λ√
1− |a|2 −λa√
1− |b|2 λab
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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}?
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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}?
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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}?
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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}?
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• 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.
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• 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.
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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 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
B(z) =z(z − a)(z − b)
(1− az)(1− bz)= λ.
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One, two and many triangles
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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.
Then look at the picture!
So Blaschke 3-ellipses and Poncelet 3-ellipses are the same.
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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.
Then look at the picture!
So Blaschke 3-ellipses and Poncelet 3-ellipses are the same.
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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.
Then look at the picture!
So Blaschke 3-ellipses and Poncelet 3-ellipses are the same.
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How this works for degree-n Blaschke products
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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.
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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}.
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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}.
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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}.
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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}.
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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.
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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
ϕaj
√1− |ak |2
1− akz, . . .}.
What’s the matrix representation for SB with respect to this basis?
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For two zeros it’s...
A =
[a√
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?
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For two zeros it’s...
A =
[a√
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?
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For two zeros it’s...
A =
[a√
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?
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The n × n matrix A is
a1√
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:
bij =
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
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Uλ =
[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 =
[x0
], ‖V tx‖ = 1,A = VUλV
t .
〈Ax , x〉 = 〈VUλVtx , x〉 = 〈UλV
tx ,V tx〉
∈W (Uλ).
One half of Halmos’s 1964 conjecture is easy!
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Uλ =
[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 =
[x0
], ‖V tx‖ = 1,A = VUλV
t .
〈Ax , x〉 = 〈VUλVtx , x〉 = 〈UλV
tx ,V tx〉 ∈W (Uλ).
One half of Halmos’s 1964 conjecture is easy!
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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〉︸ ︷︷ ︸
s
+wn+1 〈V ?1 E(n+1)(n+1)V1x , x〉︸ ︷︷ ︸
t
.
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.
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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〉︸ ︷︷ ︸
s
+wn+1 〈V ?1 E(n+1)(n+1)V1x , x〉︸ ︷︷ ︸
t
.
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.
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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〉︸ ︷︷ ︸
s
+wn+1 〈V ?1 E(n+1)(n+1)V1x , x〉︸ ︷︷ ︸
t
.
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.
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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〉︸ ︷︷ ︸
s
+wn+1 〈V ?1 E(n+1)(n+1)V1x , x〉︸ ︷︷ ︸
t
.
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.
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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〉︸ ︷︷ ︸
s
+wn+1 〈V ?1 E(n+1)(n+1)V1x , x〉︸ ︷︷ ︸
t
.
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.
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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〉︸ ︷︷ ︸
s
+wn+1 〈V ?1 E(n+1)(n+1)V1x , x〉︸ ︷︷ ︸
t
.
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.
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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〉︸ ︷︷ ︸
s
+wn+1 〈V ?1 E(n+1)(n+1)V1x , x〉︸ ︷︷ ︸
t
.
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.
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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}.
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W (SB) =⋂{W (U) : U a unitary 1-dilation of SB}.
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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).
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The advantages of this approach
You can “find” the boundary of the numerical range: Let λ ∈ T.(Dropping λ:)
Write
Fλ(z) =B(z)/z
B(z)− λ=
n∑j=1
mj
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!
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Question 2: When is the numerical range of SB elliptical?
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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.
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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.
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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.
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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.
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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.
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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.
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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 =
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!
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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 =
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.)
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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.
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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 ∈ D such that the diagonals of every circumscribing quadrilateralpass through a.
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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.
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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.
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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.
Poncelet curves
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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
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Compressions of the shift and inner functions
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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 |) <∞.
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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 ).
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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
Sθ =
[λ 00 S
]and U =
λ 0 α√
1− |λ|20 S 0
β√
1− |λ|2 0 −αβλ
.If θ(0) = 0, then λ = 0.
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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).
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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).
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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).
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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.
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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 ) =⋂{W (U) : U a unitary n − dilation of T}.
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Wilhelm Blaschke
http://www.mathe.tu-freiberg.de/fakultaet/
information/math-calendar-2016