Journal of Operator Theory

Volume 77, Issue 2, Spring 2017 pp. 455-479.

Asymmetric truncated Toeplitz operators and Toeplitz operators with matrix symbol

Authors: M. Cristina Camara (1) and Jonathan R. Partington (2)
Author institution: (1) Center for Mathematical Analysis, Geometry, and Dynamical Systems, Departamento de Matematica, Instituto Superior Tecnico, 1049-001 Lisboa, Portugal
(2) School of Mathematics, University of Leeds, Leeds LS2~9JT, U.K.

Summary: Truncated Toeplitz operators and their asymmetric versions are studied in the context of the Hardy space $H_p$ of the half-plane for $1\lt p \lt \infty$. The question of uniqueness of the symbol is solved via the characterization of the zero operator. It is shown that asymmetric truncated Toeplitz operators are equivalent after extension to $2 \times 2$ matricial Toeplitz operators, which allows one to deduce criteria for Fredholmness and invertibility. Shifted model spaces are presented in the context of invariant subspaces, allowing one to derive new Beurling-Lax theorems.

DOI: http://dx.doi.org/10.7900/jot.2016apr27.2108
Keywords: truncated Toeplitz operator, Toeplitz operator, model space, equivalence by extension, invariant subspace

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