Some Berezin number inequalities for operator matrices

TL;DR

This paper establishes new inequalities for the Berezin number of operators, especially for operator matrices, providing bounds that relate Berezin numbers to operator norms and numerical radii.

Contribution

It introduces novel Berezin number inequalities for operator matrices, expanding the theoretical understanding of Berezin symbols and their bounds.

Findings

Derived an upper bound for the Berezin number of operator matrices.

Established inequalities relating Berezin number, operator norm, and numerical radius.

Provided a specific inequality for block operator matrices involving Berezin numbers.

Abstract

The Berezin symbol $\widetilde{A}$ of an operator $A$ acting on the reproducing kernel Hilbert space ${\mathscr H}={\mathscr H(}\Omega)$ over some (non-empty) set is defined by $\widetilde{A}(\lambda)=\langle A\hat{k}_{\lambda},\hat{k}_{\lambda}\rangle\,\,\,(\lambda\in\Omega)$, where $\hat{k}_{\lambda}=\frac{{k}_{\lambda}}{\|{k}_{\lambda}\|}$ is the normalized reproducing kernel of ${\mathscr H}$. The Berezin number of operator $A$ is defined by $\mathbf{ber}(A) = \underset{\lambda \in \Omega}{\sup} \big|\tilde{A}(\lambda)\big|=\underset{\lambda \in \Omega }{\sup} \big|\langle A\hat{k}_{\lambda}, \hat{k}_{\lambda}\rangle\big|$. Moreover $\mathbf{ber}(A)\leqslant w(A)$ (numerical radius). In this paper, we present some Berezin number inequalities. Among other inequalities, it is shown that if $\mathbf{T}=\left[\begin{array}{cc} A&B C&D \end{array}\right]\in {\mathbb B}({\mathscr H(\Omega_1)}\oplus{\mathscr H(\Omega_2)})$, then \begin{align*} \mathbf{ber}(\mathbf{T}) \leqslant\frac{1}{2}\left( \mathbf{ber}(A)+ \mathbf{ber}(D)\right)+\frac{1}{2}\sqrt{\left( \mathbf{ber}(A)- \mathbf{ber}(D)\right)^2+(\|B\|+\|C\|)^2}. \end{align*}