Spectral properties of truncated Toeplitz operators by equivalence after extension

Abstract

We study truncated Toeplitz operators in model spaces K p θ for 1 < p < ∞, with essentially bounded symbols in a class including the algebra C(R ∞ ) + H + ∞ , as well as sums of analytic and anti-analytic functions satisfying a θ-separation condition, using their equivalence after extension to Toeplitz operators with 2 × 2 matrix symbols. We establish Fredholmness and invertibility criteria for truncated Toeplitz operators with θ-separated symbols and, in particular, we identify a class of operators for which semi-Fredholmness is equivalent to invertibility. For symbols in C(R ∞ ) + H + ∞ , we extend to all p ∈ (1, ∞) the spectral mapping theorem for the essential spectrum. Stronger results are obtained in the case of operators with rational symbols, or if the underlying model space is finite-dimensional.

An Iterative Wiener--Hopf Method for Triangular Matrix Functions with Exponential Factors

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Equivalence after extension and Schur coupling do not coincide on essentially incomparable Banach spaces

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