Symmetric matrix representations of truncated Toeplitz operators on finite dimensional spaces

Abstract

In this paper, we answer an open conjecture concerning complex symmetric matrices and truncated Toeplitz operators. We study matrix representations of truncated Toeplitz operators with respect to orthonormal bases which are invariant under a canonical conjugation map. In particular, we determine necessary and sufficient conditions for when a symmetric matrix is the matrix representation of a truncated Toeplitz operator with respect to a given conjugation invariant orthonormal basis. We specialise our result to the case when the conjugation invariant orthonormal basis is a modified Clark basis. With this specialisation, we answer an open conjecture in the negative, and show not every unitary equivalence between a complex symmetric matrix and a truncated Toeplitz operator arises from modified Clark basis representations. We pose a new refined conjecture for how to realise a model theory for symmetric matrices through the use of truncated Toeplitz operators, and we show this conjecture is equivalent to a specified system of polynomial equations being satisfied.