Some New Inequalities via Berezin Numbers

Year 2022, , 129 - 137, 30.06.2022

Mualla Birgül Huban , Hamdullah Başaran , Mehmet Gürdal

https://doi.org/10.47000/tjmcs.1014841

Cited By: 7

Abstract

The Berezin transform $\widetilde{T}$ and the Berezin radius of an operator
$T$ on the reproducing kernel Hilbert space $\mathcal{H}\left( Q\right) $
over some set $Q$ with the reproducing kernel $K_{\eta}$ are defined,
respectively, by
\[
\widetilde{T}(\eta)=\left\langle {T\frac{K_{\eta}}{{\left\Vert K_{\eta
}\right\Vert }},\frac{K_{\eta}}{{\left\Vert K_{\eta}\right\Vert }}%
}\right\rangle ,\ \eta\in Q\text{ and }\mathrm{ber}(T):=\sup_{\eta\in
Q}\left\vert \widetilde{T}{(\eta)}\right\vert .
\]
We study several sharp inequalities by using this bounded function
$\widetilde{T},$ involving powers of the Berezin radius and the Berezin norms
of reproducing kernel Hilbert space operators. We also give some inequalities
regarding the Berezin transforms of sum of two operators.

Keywords

Reproducing kernel Hilbert space, Berezin transform, Berezin radius, Jensen's inequaity, mixed Schwarz inequality

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