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.