Upper and lower bounds of the A-Berezin number of operators

Let $textit{A}$ be a positive bounded linear operator acting on a complex Hilbert space $mathcal{H}.$ Any positive operator $textit{A}$ induces a semiinner product on $mathcal{H}$ defined by $left langle x,yright rangle_{A}:=left langle Ax,yright rangle_{ mathcal{H}},forall x,yin mathcal{H}. For any Tin B(mathcal{H}(Omega )),$ its $textit{A}$-Berezin symbol $widetilde{T^{A}}$ is defined on $Omega$ by $widetilde{T^{A}}:= left langle Twidehat{K_{lambda }},widehat{K_{lambda }} right rangle_{A},lambdain Omega ,where widehat{K_{lambda }}$ is the normalized reproducing kernel of $mathcal{H}.$ In this paper, we introduce the notions $left ( A,r right )$-adjoint of operators and $textit{A}$-Berezin number of operators on the reproducing kernel Hilbert space and prove some upper and lower bounds of the $textit{A}$-Berezin numbers of operators. In particular, we show that $frac{1}{2}left | T right |_{A-Ber}leq maxleft { left | sin right |_{A}T,frac{sqrt{2}}{2} right }ber_{A}(T)leq ber_{A}(T),$ where $left | sin right |_{A}T$ denotes the $textit{A}$ -sinus of angle of $textit{T}$.

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