The Generalized Aluthge Transformation of Operators with Closed Range

Abstract

Let $T \in \mathcal{B}(\mathcal {H})$ be a bounded linear operator on a Hilbert space $\mathcal {H},$ with its polar decomposition give by $T = U \vert T \vert .$ The generalized Aluthge transformation of $T$ for any $\alpha , \beta >0$ is defined as $$ \Delta_{\alpha , \beta} (T)= \vert T \vert ^{\alpha}U \vert T \vert ^{\beta} .$$
This paper investigates properties of operators and their generalized Aluthge transformation, focusing on null subspaces, closed ranges, and EP conditions. For binormal operators with closed ranges, we show the generalized Aluthge transformation also has a closed range and derive a formula for its Moore-Penrose inverse. Additionally, we explore the spectrum, numerical radius, and quasinormality of the Moore-Penrose inverse.