Annals of Functional Analysis

Berezin transform of the absolute value of an operator

Namita Das and Madhusmita Sahoo

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In this article, we concentrate on the Berezin transform of the absolute value of a bounded linear operator T defined on the Bergman space La2(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 T2=T3 if and only if there exists a normal idempotent operator S on La2(D) such that ρ(T)=ρ(|S|2)=ρ(|S|2), where ρ(T) is the Berezin transform of T. We also establish that if T is compact and |Tn|=|T|n for some nN, n1, then ρ(|Tn|)=ρ(|T|n) for all nN. 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 L(La2(D)).

Article information

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

Received: 24 August 2016
Accepted: 12 March 2017
First available in Project Euclid: 17 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32A36: Bergman spaces
Secondary: 47B32: Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) [See also 46E22]

absolute value Berezin transform Bergman space positive operators reproducing kernel


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.

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