Advanced refinements of Berezin number inequalities

Year 2023, Volume: 72 Issue: 2, 386 - 396, 23.06.2023

Mehmet Gürdal Hamdullah Başaran

https://doi.org/10.31801/cfsuasmas.1160606

Cited By: 1

Abstract

For a bounded linear operator $A$ on a functional Hilbert space $\mathcal{H}\left( \Omega\right) $, with normalized reproducing kernel $\widehat {k}_{\eta}:=\frac{k_{\eta}}{\left\Vert k_{\eta}\right\Vert _{\mathcal{H}}},$ the Berezin symbol and Berezin number are defined respectively by
$\widetilde{A}\left( \eta\right) :=\left\langle A\widehat{k}_{\eta},\widehat{k}_{\eta}\right\rangle _{\mathcal{H}}$ and $\mathrm{ber}(A):=\sup_{\eta\in\Omega}\left\vert \widetilde{A}{(\eta)}\right\vert .$ A simple comparison of these properties produces the inequality $\mathrm{ber}%
\left( A\right) \leq\frac{1}{2}\left( \left\Vert A\right\Vert_{\mathrm{ber}}+\left\Vert A^{2}\right\Vert _{\mathrm{ber}}^{1/2}\right) $
(see [17]). In this paper, we prove further inequalities relating them, and also establish some inequalities for the Berezin number of operators on functional Hilbert spaces

Keywords

Berezin symbol, Berezin number, functional Hilbert space, positive operator

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