Berezin number inequalities via convex functions

Abstract

The Berezin symbol $\widetilde{A}$ of an operator $A$ on the reproducingkernel Hilbert space $\mathcal{H}\left( \Omega\right) $ over some set$\Omega$ with the reproducing kernel $\mathcal{K}_{\lambda}$ is defined by\[\tilde{A}(\lambda)=\left\langle {A\frac{\mathcal{K}_{\lambda}}{{\left\Vert\mathcal{K}_{\lambda}\right\Vert }},\frac{\mathcal{K}_{\lambda}}{{\left\Vert\mathcal{K}_{\lambda}\right\Vert }}}\right\rangle ,\ \lambda\in\Omega.\]The Berezin number of an operator $A$ is defined by\[\mathrm{ber}(A):=\sup_{\lambda\in\Omega}\left\vert \widetilde{A}{(\lambda)}\right\vert .\]We study some problems of operator theory by using this bounded function$\widetilde{A}$, including treatments of inner product inequalities via convexfunctions for the Berezin numbers of some operators. We also establish someinequalities involving of the Berezin inequalities.