Inequalities for the Berezin number of operators and related questions

Abstract

For a bounded linear operator, acting in the reproducing kernel Hilbert space ${\mathcal {H}}={\mathcal {H}}\left( \Omega \right)$ over some set $\Omega$, its Berezin symbol (or Berezin transform)$\widetilde{\text { }A}$ is defined by

$$\begin{aligned} {\widetilde{A}}\left( \lambda \right) :=\left\langle A{\widehat{k}}_{\lambda },{\widehat{k}}_{\lambda }\right\rangle ,\text { }\lambda \in \Omega , \end{aligned}$$

which is a bounded complex-valued function on $\Omega ;$ here ${\widehat{k}}_{\lambda }:=\frac{{\widehat{k}}_{\lambda }}{\left\| k_{\lambda }\right\| _{{\mathcal {H}}}}$ is the normalized reproducing kernel of ${\mathcal {H}}$. The Berezin set and the Berezin number of an operator A are defined respectively by

$$\begin{aligned} \mathrm {Ber}\left( A\right) :=\mathrm {Range}\left( {\widetilde{A}}\right) =\left\{ {\widetilde{A}}\left( \lambda \right) :\lambda \in \Omega \right\} \end{aligned}$$

and

$$\begin{aligned} \mathrm {ber}\left( A\right) :=\sup \left\{ \left| \gamma \right| :\gamma \in \mathrm {Ber}\left( A\right) \right\} =\sup _{\lambda \in \Omega }\left| {\widetilde{A}}\left( \lambda \right) \right| . \end{aligned}$$

Since $\mathrm {Ber}\left( A\right) \subset W\left( A\right)$ (numerical range) and $\mathrm {ber}\left( A\right) \le w\left( A\right)$ (numerical radius), it is natural to investigate these new numerical quantities of operators and to get some similar results as for numerical range and numerical radius. In this paper, we prove many different type inequalities, including power inequality $\mathrm {ber}\left( A^{n}\right) \le \mathrm {ber}\left( A\right) ^{n}$ for the Berezin number of operators. We also study the uncertainty principle for Berezin symbols and we describe spectrum and compactness of functions of model operator in terms of Berezin symbols. Some related problems for de Branges-Rovnyak space operators are also discussed.