Semialgebraic Dimension and Truncated Toeplitz Models for Complex Symmetric Matrices

We answer negatively a model-theoretic question for complex symmetric operators. More precisely, we show that, for every $n\geq 10$, not every $n\times n$ symmetric matrix is unitarily equivalent to a direct sum of truncated Toeplitz operators. In order to do this, we first use semialgebraic dimension, a tool from real algebraic geometry, to prove a general theorem showing that, if $\mathcal X$ is a semialgebraic family of complex symmetric matrices, then the set of complex symmetric matrices which are unitarily equivalent to an element of $\mathcal X$ is semialgebraic and has dimension at most $\dim_{\mathbb R}\mathcal X+\frac{n(n-1)}2.$ We then apply this theorem to show that when $n\geq 10$ there exist irreducible symmetric $n \times n$ matrices which are not unitarily equivalent to a truncated Toeplitz operator. Finally, we prove a positive result for a related refined representation question, which asks whether, whenever a complex symmetric matrix is unitarily equivalent to a truncated Toeplitz operator, that equivalence can be realised by a matrix representation with respect to a conjugation-invariant orthonormal basis.