On the image of the mean transform

TL;DR

This paper explores the properties of the mean transform of operators on Hilbert spaces, focusing on spectral, kernel, and image characteristics, and examines how it affects various classes of operators such as positive, normal, and unitary.

Contribution

It provides new insights into the spectral and structural properties of the mean transform and its effects on different classes of operators in operator theory.

Findings

Analyzed the spectrum of the mean transform.

Characterized the kernel and image of the mean transform.

Investigated the transformation of specific operator classes under the mean transform.

Abstract

Let $B(H)$ be the algebra of all bounded operators on a Hilbert space $H$. Let $T=V|T|$ be the polar decomposition of an operator $T\in B(H)$. The mean transform of $T$ is defined by $M(T)=\frac{T+|T|V}{2}$. In this paper, we discuss several properties related to the spectrum, the kernel, the image, the polar decomposition of mean transform. Moreover, we investigate the image and preimage by the mean transform of some class of operators as positive, normal, unitary, hyponormal and co-hyponormal operators.