We consider operators acting on a Hilbert space that can be written as the sum of a shift and a diagonal operator and determine when the operator is hyponormal. The condition is presented in terms of the norm of an explicit block Jacobi matrix whose entries are explicitly computable in many applications. We apply this result to the Toeplitz operator with symbol $z^n+c|z{|}^s$ acting on certain weighted Bergman spaces and determine for what values of the constant c this operator is hyponormal. Our result strengthens an earlier result by the second author, which in turn addressed a question posed by Fleeman and Liaw in 2019.

中文翻译：

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加权 Bergman 空间上的次规范 Toeplitz 算子

我们考虑作用于希尔伯特空间的算子，该空间可以写为移位和对角算子的总和，并确定该算子何时是次正规的。该条件根据显式块 Jacobi 矩阵的范数表示，其条目在许多应用程序中是显式可计算的。我们将此结果应用到带有符号的 Toeplitz 算子$z^n+C|z{|}^{秒}$作用于某些加权伯格曼空间，并确定该运算符对于常数c 的哪些值是次正规的。我们的结果加强了第二作者的早期结果，该结果反过来解决了 Fleeman 和 Liaw 在 2019 年提出的问题。