Semialgebraic Dimension and Truncated Toeplitz Models for Complex Symmetric Matrices
Abstract
We answer negatively a finite-dimensional unitary-model question for complex symmetric operators. More precisely, we show that, for every n≥ 10, not every n× 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 𝒳 is a semialgebraic family of complex symmetric matrices, then the set of complex symmetric matrices which are unitarily equivalent to an element of 𝒳 is semialgebraic and has dimension at most dimℝ𝒳+(n(n−1))/2. We then apply this theorem to show that when n≥ 10 there exist irreducible symmetric n × n matrices which are not unitarily equivalent to a truncated Toeplitz operator. Although unitary equivalence is too restrictive, we prove that every finite-dimensional complex symmetric operator is complex-orthogonally equivalent to a coanalytic truncated Toeplitz operator. We also answer positively a refined representation question by showing that whenever a symmetric matrix is unitarily equivalent to a truncated Toeplitz operator, it is a matrix representation of that operator with respect to a conjugation-invariant orthonormal basis.
Related
- Block Toeplitz Operators, Matrix Valued Truncated Toeplitz operators, and Conjugations
- Algebraic properties of Truncated Toeplitz + Truncated Hankel operators
- Matrix-Valued Truncated Toeplitz Operators: Properties and Applications
- Perturbations of Truncated Toeplitz operators
- Eigenvalues of Hermitian Toeplitz matrices with Fisher--Hartwig symbols
BibTeX
@misc{oloughlin2026semialgebraic,
title = {{Semialgebraic Dimension and Truncated Toeplitz Models for Complex Symmetric Matrices}},
author = {Ryan O'Loughlin},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.14019},
eprint = {2607.14019},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.14019},
note = {CC0},
}
RIS
TY - UNPB
TI - Semialgebraic Dimension and Truncated Toeplitz Models for Complex Symmetric Matrices
AU - Ryan O'Loughlin
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2607.14019
UR - https://arxiv.org/abs/2607.14019
AB - We answer negatively a finite-dimensional unitary-model question for complex symmetric operators. More precisely, we show that, for every n≥ 10, not every n× 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 𝒳 is a semialgebraic family of complex symmetric matrices, then the set of complex symmetric matrices which are unitarily equivalent to an element of 𝒳 is semialgebraic and has dimension at most dim_ℝ𝒳+(n(n−1))/2. We then apply this theorem to show that when n≥ 10 there exist irreducible symmetric n × n matrices which are not unitarily equivalent to a truncated Toeplitz operator. Although unitary equivalence is too restrictive, we prove that every finite-dimensional complex symmetric operator is complex-orthogonally equivalent to a coanalytic truncated Toeplitz operator. We also answer positively a refined representation question by showing that whenever a symmetric matrix is unitarily equivalent to a truncated Toeplitz operator, it is a matrix representation of that operator with respect to a conjugation-invariant orthonormal basis.
ER -