Spectral properties of truncated Toeplitz operators by equivalence after extension

Abstract

We study truncated Toeplitz operators in model spaces Kθp 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.