Inner functions and operator theory (original) (raw)

Inner vectors for Toeplitz operators

Complex Analysis and Spectral Theory, 2020

In this paper we survey and bring together several approaches to obtaining inner functions for Toeplitz operators. These approaches include the classical definition, the Wold decomposition, the operator-valued Poisson Integral, and Clark measures. We then extend these notions somewhat to inner functions on model spaces. Along the way we present some novel examples.

Truncated Toeplitz operators: spatial isomorphism, unitary equivalence, and similarity

Indiana University Mathematics Journal, 2010

A truncated Toeplitz operator Aϕ : K Θ → K Θ is the compression of a Toeplitz operator Tϕ : H 2 → H 2 to a model space K Θ := H 2 ΘH 2. For Θ inner, let T Θ denote the set of all bounded truncated Toeplitz operators on K Θ. Our main result is a necessary and sufficient condition on inner functions Θ 1 and Θ 2 which guarantees that T Θ 1 and T Θ 2 are spatially isomorphic. (i.e., U T Θ 1 = T Θ 2 U for some unitary U : K Θ 1 → K Θ 2). We also study operators which are unitarily equivalent to truncated Toeplitz operators and we prove that every operator on a finite dimensional Hilbert space is similar to a truncated Toeplitz operator.

On Some Problems for Toeplitz and Truncated Toeplitz Operators

For a scalar inner function �, the model space of Sz.-Nagy and Foias is the subspace K� = H2 ⊖�H2 of the classical Hardy space H2 = H2(D) over the unit disk D = {z ∈ C : |z| < 1}. For a bounded linear operator A on the model space K�, its Berezin symbol is the function e A Kdefined on D by e A K�(�) = D

Characterization of truncated Toeplitz operators by conjugations

Operators and Matrices, 2017

Truncated Toeplitz operators are C-symmetric with respect to the canonical conjugation given on an appropriate model space. However, by considering only one conjugation one cannot characterize truncated Toeplitz operators. It will be proved, for some classes of inner functions and the model spaces connected with them, that if an operator on a model space is C-symmetric for a certain family of conjugations in the model space, then is has to be truncated Toeplitz. A characterization of classical Toeplitz operators is also presented in terms of conjugations.

Operator Theory and Its Applications

2000

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Characterizations of asymmetric truncated Toeplitz operators

Banach Journal of Mathematical Analysis, 2017

The aim of this paper is to investigate asymmetric truncated Toeplitz operators with L 2 symbols between two different model spaces given by inner functions such that one divides the other. The class of symbols corresponding to the zero operator is described. Asymmetric truncated Toeplitz operators are characterized in terms of operators of rank at most two, and the relations with the corresponding symbols are studied.

Truncated Toeplitz operators on finite dimensional spaces

Operators and Matrices, 2008

In this paper, we study the matrix representations of compressions of Toeplitz operators to the finite dimensional model spaces H 2 BH 2 , where B is a finite Blaschke product. In particular, we determine necessary and sufficient conditions-in terms of the matrix representation-of when a linear transformation on H 2 BH 2 is the compression of a Toeplitz operator. This result complements a related result of Sarason [6].