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Turkish Journal of Mathematics

DOI

10.3906/mat-2108-90

Abstract

Let $A$ be a positive bounded linear operator acting on a complex Hilbert space $\mathcal{H}$. Any positive operator $A$ induces a semiinner product on $\mathcal{H}$ defined by $\left\langle x,y\right\rangle _{A}:=\left\langle Ax,y\right\rangle _{\mathcal{H}},$ $\forall x,y\in\mathcal{H}.$ For any $T\in\mathcal{B}\left( \mathcal{H}\left( \Omega\right) \right) $, its $A$-Berezin symbol $\widetilde{T}^{_{A}}$ is defined on $\Omega$ by $\widetilde{T}^{_{A}% }:=\left\langle T\widehat{K}_{\lambda},\widehat{K}_{\lambda}\right\rangle _{A},$ $\lambda\in\Omega,$where $\widehat{K}_{\lambda}$ is the normalized reproducing kernel of $\mathcal{H}$. In this paper, we introduce the notions $\left( A,r\right) $-adjoint of operators and $A$-Berezin number of operators on the reproducing kernel Hilbert space and prove some upper and lower bounds of the $A$-Berezin numbers of operators. In particular, we show that \[ \frac{1}{2}\left\Vert T\right\Vert _{A-\mathrm{Ber}}\leq\max\left\{ \left\vert \sin\right\vert _{A}T,\frac{\sqrt{2}}{2}\right\} \mathrm{ber}% _{A}\left( T\right) \leq\mathrm{ber}_{A}\left( T\right) , \] where $\left\vert \sin\right\vert _{A}T$ denotes the $A$-sinus of angle of $T$.

Keywords

Reproducing kernel Hilbert space, Berezin symbol, Berezin number, $A$-Berezin number, positive operator, semiinner product

First Page

189

Last Page

206

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