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May 2018 Berezin transform of the absolute value of an operator
Namita Das, Madhusmita Sahoo
Ann. Funct. Anal. 9(2): 151-165 (May 2018). DOI: 10.1215/20088752-2017-0035

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 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)).

Citation

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Namita Das. Madhusmita Sahoo. "Berezin transform of the absolute value of an operator." Ann. Funct. Anal. 9 (2) 151 - 165, May 2018. https://doi.org/10.1215/20088752-2017-0035

Information

Received: 24 August 2016; Accepted: 12 March 2017; Published: May 2018
First available in Project Euclid: 17 October 2017

zbMATH: 06873693
MathSciNet: MR3795081
Digital Object Identifier: 10.1215/20088752-2017-0035

Subjects:
Primary: ‎32A36‎
Secondary: 47B32

Keywords: absolute value , ‎Berezin transform , Bergman space , ‎positive operators , reproducing kernel

Rights: Copyright © 2018 Tusi Mathematical Research Group

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Vol.9 • No. 2 • May 2018
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