recent results on superoptimal approximation by …recent results on superoptimal approximation by...
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![Page 1: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/1.jpg)
Recent results on superoptimal approximation bymeromorphic functions
Alberto A. Condori
Department of MathematicsFlorida Gulf Coast University
Saturday, March 19, 2011
27th South Eastern Analysis MeetingUniversity of Florida
Alberto A. Condori Meromorphic approximation
![Page 2: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/2.jpg)
Notation
1 D is the open unit disk & T is the unit circle.
2 Mn denotes the space of n × n matrices equipped with theoperator norm ‖ · ‖Mn .
3 For an operator T and k ≥ 0, we define
sk(T ) = inf{‖T − R‖ : rank R ≤ k}
and‖T‖e = inf{‖T − K‖ : K is compact }.
4 L∞(Mn) is equipped with ‖Φ‖∞ = ess supζ∈T‖Φ(ζ)‖Mn .
Alberto A. Condori Meromorphic approximation
![Page 3: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/3.jpg)
Notation
1 D is the open unit disk & T is the unit circle.
2 Mn denotes the space of n × n matrices equipped with theoperator norm ‖ · ‖Mn .
3 For an operator T and k ≥ 0, we define
sk(T ) = inf{‖T − R‖ : rank R ≤ k}
and‖T‖e = inf{‖T − K‖ : K is compact }.
4 L∞(Mn) is equipped with ‖Φ‖∞ = ess supζ∈T‖Φ(ζ)‖Mn .
Alberto A. Condori Meromorphic approximation
![Page 4: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/4.jpg)
Notation
1 D is the open unit disk & T is the unit circle.
2 Mn denotes the space of n × n matrices equipped with theoperator norm ‖ · ‖Mn .
3 For an operator T and k ≥ 0, we define
sk(T ) = inf{‖T − R‖ : rank R ≤ k}
and‖T‖e = inf{‖T − K‖ : K is compact }.
4 L∞(Mn) is equipped with ‖Φ‖∞ = ess supζ∈T‖Φ(ζ)‖Mn .
Alberto A. Condori Meromorphic approximation
![Page 5: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/5.jpg)
Notation
1 D is the open unit disk & T is the unit circle.
2 Mn denotes the space of n × n matrices equipped with theoperator norm ‖ · ‖Mn .
3 For an operator T and k ≥ 0, we define
sk(T ) = inf{‖T − R‖ : rank R ≤ k}
and‖T‖e = inf{‖T − K‖ : K is compact }.
4 L∞(Mn) is equipped with ‖Φ‖∞ = ess supζ∈T‖Φ(ζ)‖Mn .
Alberto A. Condori Meromorphic approximation
![Page 6: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/6.jpg)
Notation
1 D is the open unit disk & T is the unit circle.
2 Mn denotes the space of n × n matrices equipped with theoperator norm ‖ · ‖Mn .
3 For an operator T and k ≥ 0, we define
sk(T ) = inf{‖T − R‖ : rank R ≤ k}
and‖T‖e = inf{‖T − K‖ : K is compact }.
4 L∞(Mn) is equipped with ‖Φ‖∞ = ess supζ∈T‖Φ(ζ)‖Mn .
Alberto A. Condori Meromorphic approximation
![Page 7: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/7.jpg)
Poles and H∞(k)(Mn)
A finite Blaschke-Potapov product of degree k is an n × nmatrix-valued function of the form
B(z) = U0
( z−a11−a1z
OO In−1
)U1 . . .Uk−1
( z−ak1−akz O
O In−1
)Uk ,
where a1, . . . , ak ∈ D and U0,U1, . . . ,Uk are constant n × nunitary matrices.
A matrix-valued function Q ∈ L∞(Mn) is said to have at most kpoles in D if there is a Blaschke-Potapov product B of degree ksuch that QB ∈ H∞(Mn).
H∞(k)(Mn) consists of matrix-valued functions Q with at most kpoles in D.
Alberto A. Condori Meromorphic approximation
![Page 8: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/8.jpg)
Poles and H∞(k)(Mn)
A finite Blaschke-Potapov product of degree k is an n × nmatrix-valued function of the form
B(z) = U0
( z−a11−a1z
OO In−1
)U1 . . .Uk−1
( z−ak1−akz O
O In−1
)Uk ,
where a1, . . . , ak ∈ D and U0,U1, . . . ,Uk are constant n × nunitary matrices.
A matrix-valued function Q ∈ L∞(Mn) is said to have at most kpoles in D if there is a Blaschke-Potapov product B of degree ksuch that QB ∈ H∞(Mn).
H∞(k)(Mn) consists of matrix-valued functions Q with at most kpoles in D.
Alberto A. Condori Meromorphic approximation
![Page 9: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/9.jpg)
Poles and H∞(k)(Mn)
A finite Blaschke-Potapov product of degree k is an n × nmatrix-valued function of the form
B(z) = U0
( z−a11−a1z
OO In−1
)U1 . . .Uk−1
( z−ak1−akz O
O In−1
)Uk ,
where a1, . . . , ak ∈ D and U0,U1, . . . ,Uk are constant n × nunitary matrices.
A matrix-valued function Q ∈ L∞(Mn) is said to have at most kpoles in D if there is a Blaschke-Potapov product B of degree ksuch that QB ∈ H∞(Mn).
H∞(k)(Mn) consists of matrix-valued functions Q with at most kpoles in D.
Alberto A. Condori Meromorphic approximation
![Page 10: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/10.jpg)
Nehari-Takagi problem
Definition
Let k ≥ 0. Given Φ ∈ L∞(Mn), we say that Q is a bestapproximation in H∞(k)(Mn) to Φ if Q has at most k poles and
‖Φ− Q‖L∞(Mn) = distL∞(Mn)(Φ,H∞(k)(Mn)).
How can we define “very best” approximation in order to obtainuniqueness?
Alberto A. Condori Meromorphic approximation
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Nehari-Takagi problem
Definition
Let k ≥ 0. Given Φ ∈ L∞(Mn), we say that Q is a bestapproximation in H∞(k)(Mn) to Φ if Q has at most k poles and
‖Φ− Q‖L∞(Mn) = distL∞(Mn)(Φ,H∞(k)(Mn)).
How can we define “very best” approximation in order to obtainuniqueness?
Alberto A. Condori Meromorphic approximation
![Page 12: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/12.jpg)
Superoptimal meromorphic approximation in H∞(k)(Mn)
Definition (Young)
Let k ≥ 0 and Φ ∈ L∞(Mn). We say that Q is a superoptimalmeromorphic approximant of Φ in H∞(k)(Mn) if Q has at most kpoles in D and minimizes the essential suprema of singular valuessj ((Φ− Q)(ζ)), j ≥ 0, with respect to the lexicographic ordering:
Q minimizes ess supζ∈T
s0(Φ(ζ)− Q(ζ)) on H∞(k)(Mn)
then... minimize ess supζ∈T
s1(Φ(ζ)− Q(ζ))
then... minimize ess supζ∈T
s2(Φ(ζ)− Q(ζ)) . . . and so on.
For j ≥ 0, the number t(k)j
def= ess supζ∈T sj(Φ(ζ)− Q(ζ)) is called
the jth superoptimal singular value of Φ of degree k.
Alberto A. Condori Meromorphic approximation
![Page 13: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/13.jpg)
Superoptimal meromorphic approximation in H∞(k)(Mn)
Definition (Young)
Let k ≥ 0 and Φ ∈ L∞(Mn). We say that Q is a superoptimalmeromorphic approximant of Φ in H∞(k)(Mn) if Q has at most kpoles in D and minimizes the essential suprema of singular valuessj ((Φ− Q)(ζ)), j ≥ 0, with respect to the lexicographic ordering:
Q minimizes ess supζ∈T
s0(Φ(ζ)− Q(ζ)) on H∞(k)(Mn)
then... minimize ess supζ∈T
s1(Φ(ζ)− Q(ζ))
then... minimize ess supζ∈T
s2(Φ(ζ)− Q(ζ)) . . . and so on.
For j ≥ 0, the number t(k)j
def= ess supζ∈T sj(Φ(ζ)− Q(ζ)) is called
the jth superoptimal singular value of Φ of degree k.
Alberto A. Condori Meromorphic approximation
![Page 14: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/14.jpg)
Superoptimal meromorphic approximation in H∞(k)(Mn)
Definition (Young)
Let k ≥ 0 and Φ ∈ L∞(Mn). We say that Q is a superoptimalmeromorphic approximant of Φ in H∞(k)(Mn) if Q has at most kpoles in D and minimizes the essential suprema of singular valuessj ((Φ− Q)(ζ)), j ≥ 0, with respect to the lexicographic ordering:
Q minimizes ess supζ∈T
s0(Φ(ζ)− Q(ζ)) on H∞(k)(Mn)
then... minimize ess supζ∈T
s1(Φ(ζ)− Q(ζ))
then... minimize ess supζ∈T
s2(Φ(ζ)− Q(ζ)) . . . and so on.
For j ≥ 0, the number t(k)j
def= ess supζ∈T sj(Φ(ζ)− Q(ζ)) is called
the jth superoptimal singular value of Φ of degree k.
Alberto A. Condori Meromorphic approximation
![Page 15: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/15.jpg)
Tools: Hankel and Toeplitz operators
Let P+ and P− := I − P+ denote the orthogonal projections fromL2(Cn) onto H2(Cn) and H2
−(Cn) = L2(Cn) H2(Cn),respectively.
Given Φ ∈ L∞(Mn), we define
1 the Toeplitz operator TΦ : H2(Cn)→ H2(Cn) by
TΦf = P+Φf for f ∈ H2(Cn),
and
2 Hankel operator HΦ : H2(Cn)→ H2−(Cn) by
HΦf = P−Φf for f ∈ H2(Cn).
Alberto A. Condori Meromorphic approximation
![Page 16: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/16.jpg)
Tools: Hankel and Toeplitz operators
Let P+ and P− := I − P+ denote the orthogonal projections fromL2(Cn) onto H2(Cn) and H2
−(Cn) = L2(Cn) H2(Cn),respectively.
Given Φ ∈ L∞(Mn), we define
1 the Toeplitz operator TΦ : H2(Cn)→ H2(Cn) by
TΦf = P+Φf for f ∈ H2(Cn),
and
2 Hankel operator HΦ : H2(Cn)→ H2−(Cn) by
HΦf = P−Φf for f ∈ H2(Cn).
Alberto A. Condori Meromorphic approximation
![Page 17: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/17.jpg)
Tools: Hankel and Toeplitz operators
Let P+ and P− := I − P+ denote the orthogonal projections fromL2(Cn) onto H2(Cn) and H2
−(Cn) = L2(Cn) H2(Cn),respectively.
Given Φ ∈ L∞(Mn), we define
1 the Toeplitz operator TΦ : H2(Cn)→ H2(Cn) by
TΦf = P+Φf for f ∈ H2(Cn),
and
2 Hankel operator HΦ : H2(Cn)→ H2−(Cn) by
HΦf = P−Φf for f ∈ H2(Cn).
Alberto A. Condori Meromorphic approximation
![Page 18: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/18.jpg)
Why are these operators useful?
Theorem (AAK-Treil)
For Φ ∈ L∞(Mn), sk(HΦ) = distL∞(Mn)(Φ,H∞(k)(Mn))
= min
{ess sup
ζ∈Ts0(Φ(ζ)− Q(ζ)) : Q ∈ H∞(k)(Mn)
}.
How about uniqueness of superoptimal approximant?
We say that Φ is k-admissible if ‖HΦ‖e is less than the smallestnon-zero superoptimal singular value of Φ of degree k .
Theorem (Peller-Young, Treil)
If Φ is k-admissible and sk(HΦ) < sk−1(HΦ), then Φ has a uniquesuperoptimal meromorphic approximant in H∞(k)(Mn) and
sj(Φ(ζ)− Q(ζ)) = t(k)j for a.e. ζ ∈ T, j ≥ 0.
Alberto A. Condori Meromorphic approximation
![Page 19: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/19.jpg)
Why are these operators useful?
Theorem (AAK-Treil)
For Φ ∈ L∞(Mn), sk(HΦ) = distL∞(Mn)(Φ,H∞(k)(Mn))
= min
{ess sup
ζ∈Ts0(Φ(ζ)− Q(ζ)) : Q ∈ H∞(k)(Mn)
}.
How about uniqueness of superoptimal approximant?
We say that Φ is k-admissible if ‖HΦ‖e is less than the smallestnon-zero superoptimal singular value of Φ of degree k .
Theorem (Peller-Young, Treil)
If Φ is k-admissible and sk(HΦ) < sk−1(HΦ), then Φ has a uniquesuperoptimal meromorphic approximant in H∞(k)(Mn) and
sj(Φ(ζ)− Q(ζ)) = t(k)j for a.e. ζ ∈ T, j ≥ 0.
Alberto A. Condori Meromorphic approximation
![Page 20: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/20.jpg)
Why are these operators useful?
Theorem (AAK-Treil)
For Φ ∈ L∞(Mn), sk(HΦ) = distL∞(Mn)(Φ,H∞(k)(Mn))
= min
{ess sup
ζ∈Ts0(Φ(ζ)− Q(ζ)) : Q ∈ H∞(k)(Mn)
}.
How about uniqueness of superoptimal approximant?
We say that Φ is k-admissible if ‖HΦ‖e is less than the smallestnon-zero superoptimal singular value of Φ of degree k .
Theorem (Peller-Young, Treil)
If Φ is k-admissible and sk(HΦ) < sk−1(HΦ), then Φ has a uniquesuperoptimal meromorphic approximant in H∞(k)(Mn) and
sj(Φ(ζ)− Q(ζ)) = t(k)j for a.e. ζ ∈ T, j ≥ 0.
Alberto A. Condori Meromorphic approximation
![Page 21: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/21.jpg)
Why are these operators useful?
Theorem (AAK-Treil)
For Φ ∈ L∞(Mn), sk(HΦ) = distL∞(Mn)(Φ,H∞(k)(Mn))
= min
{ess sup
ζ∈Ts0(Φ(ζ)− Q(ζ)) : Q ∈ H∞(k)(Mn)
}.
How about uniqueness of superoptimal approximant?
We say that Φ is k-admissible if ‖HΦ‖e is less than the smallestnon-zero superoptimal singular value of Φ of degree k .
Theorem (Peller-Young, Treil)
If Φ is k-admissible and sk(HΦ) < sk−1(HΦ), then Φ has a uniquesuperoptimal meromorphic approximant in H∞(k)(Mn) and
sj(Φ(ζ)− Q(ζ)) = t(k)j for a.e. ζ ∈ T, j ≥ 0.
Alberto A. Condori Meromorphic approximation
![Page 22: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/22.jpg)
Why are these operators useful?
Theorem (AAK-Treil)
For Φ ∈ L∞(Mn), sk(HΦ) = distL∞(Mn)(Φ,H∞(k)(Mn))
= min
{ess sup
ζ∈Ts0(Φ(ζ)− Q(ζ)) : Q ∈ H∞(k)(Mn)
}.
How about uniqueness of superoptimal approximant?
We say that Φ is k-admissible if ‖HΦ‖e is less than the smallestnon-zero superoptimal singular value of Φ of degree k .
Theorem (Peller-Young, Treil)
If Φ is k-admissible and sk(HΦ) < sk−1(HΦ), then Φ has a uniquesuperoptimal meromorphic approximant in H∞(k)(Mn) and
sj(Φ(ζ)− Q(ζ)) = t(k)j for a.e. ζ ∈ T, j ≥ 0.
Alberto A. Condori Meromorphic approximation
![Page 23: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/23.jpg)
The Toeplitz operator with symbol Φ− Q
Theorem
Suppose
1 Φ is k-admissible,
2 sk(HΦ) < sk−1(HΦ), and
3 Φ has n non-zero superoptimal singular values of degree k.
Then the Toeplitz operator TΦ−Q is Fredholm and
ind TΦ−Q = dim ker TΦ−Q .
Alberto A. Condori Meromorphic approximation
![Page 24: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/24.jpg)
The Toeplitz operator with symbol Φ− Q
Theorem
Suppose
1 Φ is k-admissible,
2 sk(HΦ) < sk−1(HΦ), and
3 Φ has n non-zero superoptimal singular values of degree k.
Then the Toeplitz operator TΦ−Q is Fredholm and
ind TΦ−Q = dim ker TΦ−Q .
Alberto A. Condori Meromorphic approximation
![Page 25: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/25.jpg)
The Toeplitz operator with symbol Φ− Q
Theorem
Suppose
1 Φ is k-admissible,
2 sk(HΦ) < sk−1(HΦ), and
3 Φ has n non-zero superoptimal singular values of degree k.
Then the Toeplitz operator TΦ−Q is Fredholm and
ind TΦ−Q = dim ker TΦ−Q .
Alberto A. Condori Meromorphic approximation
![Page 26: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/26.jpg)
The Toeplitz operator with symbol Φ− Q
Theorem
Suppose
1 Φ is k-admissible,
2 sk(HΦ) < sk−1(HΦ), and
3 Φ has n non-zero superoptimal singular values of degree k.
Then the Toeplitz operator TΦ−Q is Fredholm and
ind TΦ−Q = dim ker TΦ−Q .
Alberto A. Condori Meromorphic approximation
![Page 27: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/27.jpg)
Can we compute the index of TΦ−Q?
Question: ind TΦ−Q = 2k + µ?
Let Φ =1√2
(z5 + 1
3 z −13 z2
z4 13 z
). Then
s0(HΦ) =
√10
3, s1(HΦ) = s2(HΦ) = s3(HΦ) = 1,
s4(HΦ) =1√2, and s5(HΦ) =
1
3,
and so 2k + µ = 5, where µ is the multiplicity of s1(HΦ).
The superoptimal approximant of Φ with at most 1 pole is
Q =1√2
(13 z OO O
).
However, ind TΦ−Q = dim ker TΦ−Q = 4 even though 2k + µ = 5!
Alberto A. Condori Meromorphic approximation
![Page 28: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/28.jpg)
Can we compute the index of TΦ−Q?
Question: ind TΦ−Q = 2k + µ?
Let Φ =1√2
(z5 + 1
3 z −13 z2
z4 13 z
). Then
s0(HΦ) =
√10
3, s1(HΦ) = s2(HΦ) = s3(HΦ) = 1,
s4(HΦ) =1√2, and s5(HΦ) =
1
3,
and so 2k + µ = 5, where µ is the multiplicity of s1(HΦ).
The superoptimal approximant of Φ with at most 1 pole is
Q =1√2
(13 z OO O
).
However, ind TΦ−Q = dim ker TΦ−Q = 4 even though 2k + µ = 5!
Alberto A. Condori Meromorphic approximation
![Page 29: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/29.jpg)
Can we compute the index of TΦ−Q?
Question: ind TΦ−Q = 2k + µ?
Let Φ =1√2
(z5 + 1
3 z −13 z2
z4 13 z
). Then
s0(HΦ) =
√10
3, s1(HΦ) = s2(HΦ) = s3(HΦ) = 1,
s4(HΦ) =1√2, and s5(HΦ) =
1
3,
and so 2k + µ = 5, where µ is the multiplicity of s1(HΦ).
The superoptimal approximant of Φ with at most 1 pole is
Q =1√2
(13 z OO O
).
However, ind TΦ−Q = dim ker TΦ−Q = 4 even though 2k + µ = 5!
Alberto A. Condori Meromorphic approximation
![Page 30: Recent results on superoptimal approximation by …Recent results on superoptimal approximation by meromorphic functions Alberto A. Condori Department of Mathematics Florida Gulf Coast](https://reader034.vdocuments.mx/reader034/viewer/2022042307/5ed3020af83e157da72a01f1/html5/thumbnails/30.jpg)
Can we compute the index of TΦ−Q?
Question: ind TΦ−Q = 2k + µ?
Let Φ =1√2
(z5 + 1
3 z −13 z2
z4 13 z
). Then
s0(HΦ) =
√10
3, s1(HΦ) = s2(HΦ) = s3(HΦ) = 1,
s4(HΦ) =1√2, and s5(HΦ) =
1
3,
and so 2k + µ = 5, where µ is the multiplicity of s1(HΦ).
The superoptimal approximant of Φ with at most 1 pole is
Q =1√2
(13 z OO O
).
However, ind TΦ−Q = dim ker TΦ−Q = 4 even though 2k + µ = 5!
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A special subspace
Let B and Λ be Blaschke-Potapov products such that
ker HQ = BH2(Cn) and ker HQt = ΛH2(Cn).
LetE = {ξ ∈ ker HQ : ‖HΦξ‖2 = ‖(Φ− Q)ξ‖2}
andU = Λt(Φ− Q)B.
Theorem
If the number of superoptimal singular values of Φ of degree kequals n, then
1 E = B ker TU
2 the Toeplitz operator TU is Fredholm and
3 ind TU = dim ker TU ≥ n.
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A special subspace
Let B and Λ be Blaschke-Potapov products such that
ker HQ = BH2(Cn) and ker HQt = ΛH2(Cn).
LetE = {ξ ∈ ker HQ : ‖HΦξ‖2 = ‖(Φ− Q)ξ‖2}
andU = Λt(Φ− Q)B.
Theorem
If the number of superoptimal singular values of Φ of degree kequals n, then
1 E = B ker TU
2 the Toeplitz operator TU is Fredholm and
3 ind TU = dim ker TU ≥ n.
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A special subspace
Let B and Λ be Blaschke-Potapov products such that
ker HQ = BH2(Cn) and ker HQt = ΛH2(Cn).
LetE = {ξ ∈ ker HQ : ‖HΦξ‖2 = ‖(Φ− Q)ξ‖2}
andU = Λt(Φ− Q)B.
Theorem
If the number of superoptimal singular values of Φ of degree kequals n, then
1 E = B ker TU
2 the Toeplitz operator TU is Fredholm and
3 ind TU = dim ker TU ≥ n.
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A special subspace
Let B and Λ be Blaschke-Potapov products such that
ker HQ = BH2(Cn) and ker HQt = ΛH2(Cn).
LetE = {ξ ∈ ker HQ : ‖HΦξ‖2 = ‖(Φ− Q)ξ‖2}
andU = Λt(Φ− Q)B.
Theorem
If the number of superoptimal singular values of Φ of degree kequals n, then
1 E = B ker TU
2 the Toeplitz operator TU is Fredholm and
3 ind TU = dim ker TU ≥ n.
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A special subspace
Let B and Λ be Blaschke-Potapov products such that
ker HQ = BH2(Cn) and ker HQt = ΛH2(Cn).
LetE = {ξ ∈ ker HQ : ‖HΦξ‖2 = ‖(Φ− Q)ξ‖2}
andU = Λt(Φ− Q)B.
Theorem
If the number of superoptimal singular values of Φ of degree kequals n, then
1 E = B ker TU
2 the Toeplitz operator TU is Fredholm and
3 ind TU = dim ker TU ≥ n.
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The index formula
Theorem
Let E = {ξ ∈ ker HQ : ‖HΦξ‖2 = ‖(Φ− Q)ξ‖2}. Then theToeplitz operator TΦ−Q is Fredholm and has index
ind TΦ−Q = 2k + dim E .
In particular, dim ker TΦ−Q ≥ 2k + n.
Corollary
If all superoptimal singular values of degree k of Φ are equal, then
ind TΦ−Q = dim ker TΦ−Q = 2k + µ
holds, where µ denotes the multiplicity of the singular valuesk(HΦ).
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The index formula
Theorem
Let E = {ξ ∈ ker HQ : ‖HΦξ‖2 = ‖(Φ− Q)ξ‖2}. Then theToeplitz operator TΦ−Q is Fredholm and has index
ind TΦ−Q = 2k + dim E .
In particular, dim ker TΦ−Q ≥ 2k + n.
Corollary
If all superoptimal singular values of degree k of Φ are equal, then
ind TΦ−Q = dim ker TΦ−Q = 2k + µ
holds, where µ denotes the multiplicity of the singular valuesk(HΦ).
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Open problem #1
Sharp estimates on the “degree” of Q:
Theorem (Peller-Vasyunin)
If Φ is a rational function 2× 2 with poles off T, then “generically”the best analytic approximant Q to ϕ is a rational function and
deg Q ≤ deg Φ− 2 unless Φ ∈ H∞(M2).
In general, one has
deg Q ≤ 2 deg Φ− 3
and this inequality is sharp!
#1. What can be said for matrix-valued functions of arbitrary size?
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Open problem #1
Sharp estimates on the “degree” of Q:
Theorem (Peller-Vasyunin)
If Φ is a rational function 2× 2 with poles off T, then “generically”the best analytic approximant Q to ϕ is a rational function and
deg Q ≤ deg Φ− 2 unless Φ ∈ H∞(M2).
In general, one has
deg Q ≤ 2 deg Φ− 3
and this inequality is sharp!
#1. What can be said for matrix-valued functions of arbitrary size?
Alberto A. Condori Meromorphic approximation
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Open problem #1
Sharp estimates on the “degree” of Q:
Theorem (Peller-Vasyunin)
If Φ is a rational function 2× 2 with poles off T, then “generically”the best analytic approximant Q to ϕ is a rational function and
deg Q ≤ deg Φ− 2 unless Φ ∈ H∞(M2).
In general, one has
deg Q ≤ 2 deg Φ− 3
and this inequality is sharp!
#1. What can be said for matrix-valued functions of arbitrary size?
Alberto A. Condori Meromorphic approximation
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Open problem #1
Sharp estimates on the “degree” of Q:
Theorem (Peller-Vasyunin)
If Φ is a rational function 2× 2 with poles off T, then “generically”the best analytic approximant Q to ϕ is a rational function and
deg Q ≤ deg Φ− 2 unless Φ ∈ H∞(M2).
In general, one has
deg Q ≤ 2 deg Φ− 3
and this inequality is sharp!
#1. What can be said for matrix-valued functions of arbitrary size?
Alberto A. Condori Meromorphic approximation
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Open problem #2 & # 3
#2. How can we verify that a matrix-valued function Φ ∈ L∞ hasn non-zero superoptimal singular values?
#3. Find a characterization for the superoptimal approximant.
Thank you!
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Open problem #2 & # 3
#2. How can we verify that a matrix-valued function Φ ∈ L∞ hasn non-zero superoptimal singular values?
#3. Find a characterization for the superoptimal approximant.
Thank you!
Alberto A. Condori Meromorphic approximation
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Open problem #2 & # 3
#2. How can we verify that a matrix-valued function Φ ∈ L∞ hasn non-zero superoptimal singular values?
#3. Find a characterization for the superoptimal approximant.
Thank you!
Alberto A. Condori Meromorphic approximation