## Annals of Functional Analysis

### 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 $L_{a}^{2}(\mathbb{D})$ of the open unit disk. We establish some sufficient conditions on $T$ which guarantee that the Berezin transform of $|T|$ majorizes the Berezin transform of $|T^{*}|$. 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 $L_{a}^{2}(\mathbb{D})$ such that $\rho(T)=\rho(|S|^{2})=\rho(|S^{*}|^{2})$, where $\rho(T)$ is the Berezin transform of $T$. We also establish that if $T$ is compact and $|T^{n}|=|T|^{n}$ for some $n\in\mathbb{N}$, $n\neq1$, then $\rho(|T^{n}|)=\rho(|T|^{n})$ for all $n\in\mathbb{N}$. Further, if $T=U|T|$ 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 $\mathcal{L}(L_{a}^{2}(\mathbb{D}))$.

#### Article information

Source
Ann. Funct. Anal., Volume 9, Number 2 (2018), 151-165.

Dates
Accepted: 12 March 2017
First available in Project Euclid: 17 October 2017

https://projecteuclid.org/euclid.afa/1508205626

Digital Object Identifier
doi:10.1215/20088752-2017-0035

Mathematical Reviews number (MathSciNet)
MR3795081

Zentralblatt MATH identifier
06873693

#### Citation

Das, Namita; Sahoo, Madhusmita. Berezin transform of the absolute value of an operator. Ann. Funct. Anal. 9 (2018), no. 2, 151--165. doi:10.1215/20088752-2017-0035. https://projecteuclid.org/euclid.afa/1508205626

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