Reverse inequalities for the Berezin number of operators

TL;DR

This paper explores reverse inequalities for the Berezin number of operators on reproducing kernel Hilbert spaces, refining existing bounds and establishing new inequalities that relate Berezin number, norm, and Berezin norm.

Contribution

It introduces new reverse inequalities for the Berezin number, enhancing the understanding of operator characteristics in reproducing kernel Hilbert spaces.

Findings

Refined the inequality: $ber(A) \\leq ( \\|A\|_{ber}^2 - \\inf_{\\lambda} \\lVert (A - \\widetilde{A}(\\lambda)) \\widehat{k}_{\\lambda} \\rVert^2 )^{1/2}$

Established relationships between Berezin number, Berezin norm, and operator norm

Provided bounds that improve upon classical inequalities in operator theory.

Abstract

For a bounded linear operator $A$ on a reproducing kernel Hilbert space $\mathcal{H}(\Omega)$, with normalized reproducing kernel $\widehat{k}_{\lambda} = \frac{k_{\lambda}}{\lVert k_{\lambda}\lVert}$, the Berezin symbol, Berezin number and Berezin norm are defined respectively by $\widetilde{A}(\lambda) = \langle A\widehat{k}_{\lambda},\widehat{k}_{\lambda}\rangle$, $ber(A) = \sup_{\lambda\in\Omega}\left|\widetilde{A}(\lambda)\right|$ and $\left\|A\right\|_{ber} = \sup_{\lambda\in\Omega}\left\|A\widehat{k}_{\lambda}\right\|$. A straightforward comparison between these characteristics yields the inequalities $ber(A)\leq\left\|A\right\|_{ber}\leq\lVert A\lVert$. In this paper, we prove further inequalities relating them, and give special care to the corresponding reverse inequalities. In particular, we refine the first one of the above inequalities, namely we prove that \ $ber(A)\leq\left( \left\|A\right\|_{ber}^{2}-\inf_{\lambda\in\Omega}\left\lVert (A-\widetilde{A}(\lambda))\widehat{k}_{\lambda}\right\lVert^{2}\right) ^{\frac{1}{2}}$.