Some New Inequalities via Berezin Numbers

Authors : Mualla Birgül HUBAN, Hamdullah BAŞARAN, Mehmet GÜRDAL

Pages : 129-137

Doi:10.47000/tjmcs.1014841

View : 37 | Download : 11

Publication Date : 2022-06-30

Article Type : Research Paper

Abstract :The Berezin transform $\widetilde{T}$ and the Berezin radius of an operator $T$ on the reproducing kernel Hilbert space $\mathcal{H}\leftinsert ignore into journalissuearticles values( Q\right); $ over some set $Q$ with the reproducing kernel $K_{\eta}$ are defined, respectively, by \[ \widetilde{T}insert ignore into journalissuearticles values(\eta);=\left\langle {T\frac{K_{\eta}}{{\left\Vert K_{\eta }\right\Vert }},\frac{K_{\eta}}{{\left\Vert K_{\eta}\right\Vert }}% }\right\rangle ,\ \eta\in Q\text{ and }\mathrm{ber}insert ignore into journalissuearticles values(T);:=\sup_{\eta\in Q}\left\vert \widetilde{T}{insert ignore into journalissuearticles values(\eta);}\right\vert . \] 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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