发表机构
Department of Mathematics and Statistics, University of Reading（雷丁大学数学与统计系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究复对称矩阵相关模型理论问题，利用半代数维数证明定理，表明\(n\geq10\)时存在不可约对称\(n\times n\)矩阵不酉等价于截断Toeplitz算子，还对相关精细表示问题给出肯定结果。
AI 中文摘要
我们对复对称算子的一个模型理论问题给出了否定回答。具体而言，我们证明对于每个\(n\geq10\)，并非每个\(n\times n\)对称矩阵都酉等价于截断Toeplitz算子的直和。为此，我们首先使用实代数几何中的工具半代数维数证明一个一般定理，表明若\(\mathcal X\)是复对称矩阵的半代数族，那么与\(\mathcal X\)中元素酉等价的复对称矩阵集是半代数的且维数至多为\(\dim_{\mathbb R}\mathcal X+\frac{n(n - 1)}2\)。接着应用此定理表明当\(n\geq10\)时存在不可约对称\(n\times n\)矩阵，其不酉等价于截断Toeplitz算子。最后，我们对一个相关的精细表示问题证明了一个肯定结果，该问题询问当一个复对称矩阵酉等价于截断Toeplitz算子时，这种等价是否能通过关于共轭不变正交基的矩阵表示来实现。
英文摘要
We answer negatively a finite-dimensional unitary-model 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. 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.
