A class of operators with normal Aluthge transformation

TL;DR

This paper demonstrates that generalized Aluthge transformations of weighted conditional operators are normal, leading to new insights on operator properties and conditions for equality in inequalities, with practical examples included.

Contribution

It establishes that the Aluthge transformation of a broad class of weighted conditional operators results in normal operators, a novel finding in operator theory.

Findings

Generalized Aluthge transformations of weighted conditional operators are normal.

Invertible weighted conditional operators are shown to be normal.

Conditions under which the Holder inequality becomes equality are identified.

Abstract

In this paper, we show that the generalized Aluthge transforma- tions of a large class of operators (weighted conditional type operators) are normal. As a consequence, the operator MwEMu is p-hyponormal if and only if it is normal, and under a weak condition, if MwEMu is normal, then the Holder inequality turn into equality for w, u. Also, we conclude that, invertible weighted conditional type operators are normal. In the end we give some appli- cations of p-hyponormal weighted conditional type operators. In the end, some examples are provided to illustrate concrete application of the main results of the paper.