Symmetric matrix representations of truncated Toeplitz operators on finite dimensional spaces

In this paper, we answer an open conjecture concerning com-plex symmetric matrices and truncated Toeplitz operators. We study matrix representations of truncated Toeplitz oper-ators with respect to orthonormal bases which are invariant under a canonical conjugation map. In particular, we deter-mine 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 or-thonormal 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 be-tween 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 spec-ified system of polynomial equations being satisfied.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).