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Toeplitz Matrices

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Proof. As usual, by factoring out the Blaschke product we may assume that G(z) 0 for all z D and that, therefore, there exists an analytic function h in D with G(z) exp h(z). Since G(z) exp(Re...

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Proof. As usual, by factoring out the Blaschke product we may assume that G(z) 0 for all z D and that, therefore, there exists an analytic function h in D with G(z) exp h(z). Since G(z) exp(Re...

The Shift Operators

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is the intersection of all subspaces containing S. It is obvious that ! S is always a subspace. Definition . If A is an operator and M is a subspace, we say that M is an invariant subspace of A if AM...

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is the intersection of all subspaces containing S. It is obvious that ! S is always a subspace. Definition . If A is an operator and M is a subspace, we say that M is an invariant subspace of A if AM...

The M ¿untzGSz asz Theorem

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Theorem . For each z0 D, the function (z) z0 z 1 z0z is an inner function and Mz0 {f H : f (z0) 0} H2. Proof. The function is clearly in H. Moreover, it is continuous on the closure of D....

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Theorem . For each z0 D, the function (z) z0 z 1 z0z is an inner function and Mz0 {f H : f (z0) 0} H2. Proof. The function is clearly in H. Moreover, it is continuous on the closure of D....

The HardyGHilbert Space

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Graduate Texts in Mathematics 237 Editorial Board S. Axler K.A. Ribet Graduate Texts in Mathematics 1 2 TAKEUTI/ZARING. Introduction to Axiomatic Set Theory. 2nd ed. OXTOBY. Measure and...

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Graduate Texts in Mathematics 237 Editorial Board S. Axler K.A. Ribet Graduate Texts in Mathematics 1 2 TAKEUTI/ZARING. Introduction to Axiomatic Set Theory. 2nd ed. OXTOBY. Measure and...

Spectral Structure

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98 3 Toeplitz Operators (M )n 2 1 2 2 0 1 ((ei ) )n(ei )2 d 2 En 1 ((ei ) )2 d Also, n2 m(En). n 2 1 2 2 0 n(ei )2 d m(En) 0. Thus, if we dene fn n/ n , then {fn} is a sequence...

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98 3 Toeplitz Operators (M )n 2 1 2 2 0 1 ((ei ) )n(ei )2 d 2 En 1 ((ei ) )2 d Also, n2 m(En). n 2 1 2 2 0 n(ei )2 d m(En) 0. Thus, if we dene fn n/ n , then {fn} is a sequence...

Some Facts from Functional Analysis

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As is well known and easily veried, 4 d 0 2s converges when 2s 1, and thus when s 1 . We must now estimate 1 d 2 Since cos 0 for [ , ], (1 r2 2r cos )s . 1 r2 2r cos 1 r2 for...

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As is well known and easily veried, 4 d 0 2s converges when 2s 1, and thus when s 1 . We must now estimate 1 d 2 Since cos 0 for [ , ], (1 r2 2r cos )s . 1 r2 2r cos 1 r2 for...

Singular Inner Functions

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The product n rk converges if and only if there exists r 0 such that n lim n n rk r. k1 By continuity of log on (0, 1], this occurs if and only if or, equivalently, lim n n log n rk log r, k1 lim...

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The product n rk converges if and only if there exists r 0 such that n lim n n rk r. k1 By continuity of log on (0, 1], this occurs if and only if or, equivalently, lim n n log n rk log r, k1 lim...

Self Adjointness and Normality of Hankel Operators

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M . Restricting to H and multiplying on the left by P gives H PJM f2 . Clearly, H . 4.2 Hankel Operators of Finite Rank There are many Hankel operators of nite rank. For...

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M . Restricting to H and multiplying on the left by P gives H PJM f2 . Clearly, H . 4.2 Hankel Operators of Finite Rank There are many Hankel operators of nite rank. For...

Relations Between Hankel and Toeplitz Operators

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The Hankel operators whose matrices with respect to the standard basis for H have only a nite number of entries dierent from 0 can be described in very explicitly. Note that H has this property...

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The Hankel operators whose matrices with respect to the standard basis for H have only a nite number of entries dierent from 0 can be described in very explicitly. Note that H has this property...

Invariant and Reducing Subspaces

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32 1 Introduction . Assume that there is a bounded operator A with the following property: there exist subspaces N and M invariant...

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32 1 Introduction . Assume that there is a bounded operator A with the following property: there exist subspaces N and M invariant...

Inner and Outer Functions

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1.2 Some Facts from Functional Analysis 29 where A1 is an operator on M and A4 is an operator on M . This matrix representation shows why the word reducing is...

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1.2 Some Facts from Functional Analysis 29 where A1 is an operator on M and A4 is an operator on M . This matrix representation shows why the word reducing is...

Hankel Operators of Finite Rank

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4.1 Bounded Hankel Operators 127 Theorem (Douglass Theorem). Let H, K, and L be Hilbert spaces and suppose that E : H K and F : H L are bounded...

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4.1 Bounded Hankel Operators 127 Theorem (Douglass Theorem). Let H, K, and L be Hilbert spaces and suppose that E : H K and F : H L are bounded...

Fundamental Properties of Composition Operators

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154 4 Hankel Operators T Hei Hei T Hei Hdei . 2 , it follows that Hdei 0, and thus that Since...

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154 4 Hankel Operators T Hei Hei T Hei Hdei . 2 , it follows that Hdei 0, and thus that Since...

Eigenvalues and Eigenvectors

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166 5 Composition Operators Therefore 2 1 f...

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166 5 Composition Operators Therefore 2 1 f...

Composition Operators Induced by Disk Automorphisms

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5.1 Fundamental Properties of Composition Operators 169 Composition operators are characterized as those operators whose adjoints map the set of reproducing kernels into...

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5.1 Fundamental Properties of Composition Operators 169 Composition operators are characterized as those operators whose adjoints map the set of reproducing kernels into...

Compact Hankel Operators

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134 4 Hankel Operators as as1 a1 a0 a1 a2...

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134 4 Hankel Operators as as1 a1 a0 a1 a2...

Bounded Hankel Operators

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3.3 Spectral Structure 113 and gf in L1 (S 1 ) (recall that the product of two functions in an L2 space is in L1 ; see [47, p. 66]) have...

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3.3 Spectral Structure 113 and gf in L1 (S 1 ) (recall that the product of two functions in an L2 space is in L1 ; see [47, p. 66]) have...

Blaschke Products

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2.1 The Shift Operators 41 as n approaches . Every vector fn in 2 corresponds to the vector gn in 2 (Z) whose coordinates in positions 0, 1, 2,...

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2.1 The Shift Operators 41 as n approaches . Every vector fn in 2 corresponds to the vector gn in 2 (Z) whose coordinates in positions 0, 1, 2,...

Basic Properties of Toeplitz Operators

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88 2 The Unilateral Shift and Factorization of Functions Now we can nally prove that functions of the above form are outer. Theorem . If f is in H 2 and F is dened by 2...

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88 2 The Unilateral Shift and Factorization of Functions Now we can nally prove that functions of the above form are outer. Theorem . If f is in H 2 and F is dened by 2...

Technical results

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2.1 Schur multipliers and Pellers theorem 17 and only if there exist a probability space (, ) and bounded operators T1 : L1 () L (; ), T2 : L1 (; ) L () such...

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2.1 Schur multipliers and Pellers theorem 17 and only if there exist a probability space (, ) and bounded operators T1 : L1 () L (; ), T2 : L1 (; ) L () such...

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