Characterization of truncated Toeplitz operators by conjugations

Abstract

ABSTRACT. 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.

Asymmetric truncated toeplitz operators and conjugations

“…Let H be a complex Hilbert space, and denote by L(H) the algebra of all bounded linear operators on H. A conjugation on H is an antilinear involution C : H → H such that Cf, Cg = g, f for all f, g ∈ H. Conjugations and their relations with various classes of operators have been studied in Hilbert spaces for many years. A new motivation to study them came from [7], and many interesting results have recently appeared on this topic [2,8,10,11,12,16]. In particular, the study of C-symmetric operators, i.e., operators A ∈ L(H) such that CAC = A * , has attracted much attention, with particular emphasis on the case where the underlying Hilbert spaces are model spaces, defined as follows.…”