论文标题
在有限维空间上截短的Toeplitz运算符的对称矩阵表示
Symmetric matrix representations of truncated Toeplitz operators on finite dimensional spaces
论文作者
论文摘要
在本文中，我们研究了截短的toeplitz操作员的矩阵表示，相对于正统基础，在规范共轭图下是不变的。特别是，我们确定了何时3 x-3对称矩阵是相对于给定共轭不变的正顺式基础的截断toeplitz操作员的矩阵表示。我们将结果专注于结合不变的正常基础是修改的克拉克基础。作为这种专业化的必然性，我们回答了先前所述的开放猜想，并表明并非复杂的对称矩阵与截断的toeplitz operator之间的每个单一等价都来自修改后的Clark基础表示。最后，我们表明，给定的3 x-3对称矩阵是相对于共轭不变的正常基础的截断的toeplitz operator的矩阵表示，并且仅当指定的多项式方程系统与真实解决方案满意时。
In this paper, we study matrix representations of truncated Toeplitz operators with respect to orthonormal bases which are invariant under a canonical conjugation map. In particular, we determine necessary and sufficient conditions for when a 3-by-3 symmetric matrix is the matrix representation of a truncated Toeplitz operator with respect to a given conjugation invariant orthonormal basis. We specialise our result to the case when the conjugation invariant orthonormal basis is a modified Clark basis. As a corollary to this specialisation, we answer a previously stated open conjecture in the negative, and show that not every unitary equivalence between a complex symmetric matrix and a truncated Toeplitz operator arises from a modified Clark basis representation. Finally, we show that a given 3-by-3 symmetric matrix is the matrix representation of a truncated Toeplitz operator with respect to a conjugation invariant orthonormal basis if and only if a specified system of polynomial equations is satisfied with a real solution.
