Berezin transform of the absolute value of an operator

Abstract

In this article, we concentrate on the Berezin transform of the absolute value of a bounded linear operator $T$ defined on the Bergman space $L_a^2\left(\mathbb{D}\right)$ of the open unit disk. We establish some sufficient conditions on $T$ which guarantee that the Berezin transform of $\left|T\right|$ majorizes the Berezin transform of $\left|T^{\ast }\right|$. We have shown that $T$ is self-adjoint and $T^2=T^3$ if and only if there exists a normal idempotent operator $S$ on $L_a^2\left(\mathbb{D}\right)$ such that $\rho \left(T\right)=\rho \left(\right|S{|}^2)=\rho (\left|S^{\ast }{|}^2\right)$, where $\rho \left(T\right)$ is the Berezin transform of $T$. We also establish that if $T$ is compact and $\left|T^n\right|=|T{|}^n$ for some $n\in \mathbb{N}$, $n\ne 1$, then $\rho \left(\right|T^n\left|\right)=\rho \left(\right|T{|}^n)$ for all $n\in \mathbb{N}$. Further, if $T=U\left|T\right|$ is the polar decomposition of $T$, then we present necessary and sufficient conditions on $T$ such that $|T{|}^{1/2}$ intertwines with $U$ and a contraction $X$ belonging to $\mathcal{L}\left(L_a^2\right(\mathbb{D}\left)\right)$.