On the image of the mean transform

Abstract

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