Some relationships between an operator and its transform $S_{r}(T)$

IF 0.8 Q2 MATHEMATICS Advances in Operator Theory Pub Date : 2024-02-22 DOI:10.1007/s43036-024-00317-w

Safa Menkad, Sohir Zid

引用次数: 0

Abstract

Let $ T \in \mathcal {B}(\mathcal {H})$ be a bounded linear operator on a Hilbert space $ \mathcal {H}$, and let $ T = U \vert T \vert $ be the polar decomposition of T. For any $r > 0$, the transform $S_{r}(T)$ is defined by $S_{r}(T) = U \vert T \vert ^{r} U$. In this paper, we discuss the transform $S_{r}(T)$ of some classes of operators such as p-hyponormal and rank one operators. We provide a new characterization of invertible normal operators via this transform. Afterwards, we investigate when an operator T and its transform $ S_{r}(T) $ both have closed ranges, and show that this transform preserves the class of EP operators. Finally, we present some relationships between an EP operator T, its transform $ S_{r}(T)$ and the Moore–Penrose inverse $ T^{+} $.

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算子与其变换 $$S_{r}(T)$$ 之间的一些关系

让$ T \in \mathcal {B}(\mathcal {H})$是希尔伯特空间$ \mathcal {H}$上的有界线性算子，并让$ T = U \vert T \vert $是T的极性分解。对于任意的（r >0），变换（S_{r}(T)\）的定义是（S_{r}(T) = U \vert T \vert ^{r}）。U\).本文讨论了一些类算子的变换（S_{r}(T)\)，如 p-hyponormal 算子和一阶算子。我们通过这个变换提供了可逆正则算子的新特征。之后，我们研究了当算子 T 及其变换 $ S_{r}(T) $ 都具有封闭范围时的情况，并证明这个变换保留了 EP 算子类。最后，我们介绍了 EP 算子 T、它的变换 $ S_{r}(T)$ 和摩尔-彭罗斯逆 $ T^{+} $ 之间的一些关系。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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