Some New Applications of Berezin Symbols

Abstract

We study some problems of operator theory by using Berezin symbols approach. Namely, we investigate in terms of Berezin symbols invariant subspaces of isometric composition operators on ${\mathcal {H}}\left( \Omega \right) .$ We discuss operator corona problem, in particular, the Toeplitz corona problem. Further, we characterize unitary operators in terms of Berezin symbols. We show that the well known inequality $w\left( A\right) \ge \frac{1}{2}\left\| A\right\|$ for numerical radius is not true for the Berezin number of operators, which is defined by ${\textrm{ber}}\left( A\right) :=\sup _{\lambda \in \Omega }\left| {\tilde{A}}\left( \lambda \right) \right| ,$ where ${\tilde{A}}\left( \lambda \right) :=\left\langle A\hat{k}_{\lambda },{\hat{k}}_{\lambda }\right\rangle$ is the Berezin symbol of operator $A:{\mathcal {H}}\left( \Omega \right) \rightarrow {\mathcal {H}}\left( \Omega \right) .$ Finally, we provide a lower bound for ${\textrm{ber}}\left( A\right) .$