Czechoslovak Mathematical Journal, first online, pp. 1-13


Some Berezin number inequalities for operator matrices

Mojtaba Bakherad

Received February 3, 2017.   First published February 1, 2018.

Abstract:  The Berezin symbol $\tilde{A}$ of an operator $A$ acting on the reproducing kernel Hilbert space ${\mathcal H}={\mathcal H}(\Omega)$ over some (nonempty) set is defined by $\tilde{A}(\lambda)=\langle A\hat{k}_{\lambda},\hat{k}_{\lambda}\rangle,$ $\lambda\in\Omega$, where $\hat{k}_{\lambda}={k_{\lambda}}/{\|k_{\lambda}\|}$ is the normalized reproducing kernel of ${\mathcal H}$. The Berezin number of the operator $A$ is defined by $ ber(A)=\sup_{\lambda\in\Omega}|\tilde{A}(\lambda)|=\sup_{\lambda\in\Omega}|\langle A\hat{k}_{\lambda},\hat{k}_{\lambda}\rangle|$. Moreover, $ ber(A)\leq w(A)$ (numerical radius). We present some Berezin number inequalities. Among other inequalities, it is shown that if $ T=\left[\smallmatrix A&B C&D \right]\in{\mathbb B}({\mathcal H(\Omega_1)}\oplus{\mathcal H(\Omega_2)})$, then ber( T) \leq\frac12( ber(A)+ ber(D))+\frac12\sqrt{( ber(A)- ber(D))^2+(\|B\|+\|C\|)^2}.
Keywords:  reproducing kernel; Berezin number; numerical radius; operator matrix
Classification MSC:  47A30, 15A60, 30E20, 47A12, 47B15, 47B20
DOI:  10.21136/CMJ.2018.0048-17

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Affiliations:   Mojtaba Bakherad, Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, University Blvd., P. O. Box 98155-987, Zahedan, Iran, e-mail: bakherad@member.ams.org


 
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