Spectral properties of truncated Toeplitz operators by equivalence after extension

We study truncated Toeplitz operators in model spaces K-theta(p) for 1 < p < infinity, with essentially bounded symbols in a class including the algebra C(R-infinity) H-infinity(+), as well as sums of analytic and anti-analytic functions satisfying a theta-separation condition, using their equivalence after extension to Toeplitz operators with 2 x 2 matrix symbols. We establish Fredholmness and invertibility criteria for truncated Toeplitz operators with 0-separated symbols and, in particular, we identify a class of operators for which semi-Fredholmness is equivalent to invertibility. For symbols in C(R-infinity) H-infinity(+), we extend to all p is an element of (1, infinity) 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. (C) 2015 Elsevier Inc. All rights reserved.