This paper concerns the analysis of an unbounded Toeplitz-like operator generated by a rational matrix function having poles on the unit circle \({\mathbb T}\). It extends the analysis of such operators generated by scalar rational functions with poles on \({\mathbb T}\) found in Groenewald et al. (Oper Theory Adv Appl 271:239–268, 2018; Oper Theory Adv Appl 272:133–154, 2019; Integr Equ Oper Theory 91, 2019). A Wiener–Hopf type factorization of rational matrix functions with poles and zeroes on \({\mathbb T}\) is proved and then used to analyze the Fredholm properties of such Toeplitz-like operators. A formula for the index, based on the factorization, is given. Furthermore, it is shown that the determinant of the matrix function having no zeroes on \({\mathbb T}\) is not sufficient for the Toeplitz-like operator to be Fredholm, in contrast to the classical case.

本文涉及对由在单位圆上具有极点的有理矩阵函数生成的无界Toeplitz算子的分析。（{\ mathbb T} \）。它扩展了对由标量有理函数生成的此类算符的分析，并带有Groenewald等人的\（{\ mathbb T} \）上的极点。（Oper Theory Adv Appl 271：239-268，2018; Oper Theory Adv Appl 272：133-154，2019; Integr Equ Oper Theory 91，2019）。证明了在\（{\ mathbb T} \）上具有极点和零点的有理矩阵函数的Wiener-Hopf型分解，然后将其用于分析此类Toeplitz算子的Fredholm性质。给出了基于分解的索引公式。此外，表明矩阵函数的行列式上不存在零与经典情况相反，对于类似Toeplitz的运算符来说，Fredholm不能使用\（{\ mathbb T} \）。