Semialgebraic Dimension and Truncated Toeplitz Models for Complex Symmetric Matrices

Abstract

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 $n\ge 10$ 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.

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