REVERSE INEQUALITIES FOR THE BEREZIN NUMBER OF OPERATORS

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