Berezin number and Berezin norm inequalities for operator matrices

TL;DR

This paper introduces refined upper bounds for Berezin number and Berezin norm of operator matrices, enhancing existing inequalities and providing practical estimation examples on Hardy-Hilbert spaces.

Contribution

It presents new, sharper bounds for Berezin number and norm of operator matrices, extending previous results with explicit formulas and examples.

Findings

Established bounds: A_{ber} \u2264 [A_{ij}_{ber}] and ber(A)  w([a_{ij}])

Derived bounds involve diagonal and off-diagonal entries with specific formulas

Provided examples on Hardy-Hilbert space for Berezin number and norm estimation.

Abstract

We establish new upper bounds for Berezin number and Berezin norm of operator matrices, which are refinements of the existing bounds. Among other bounds, we prove that if $A=[A_{ij}]$ is an $n\times n$ operator matrix with $A_{ij}\in\mathbb{B}(\mathcal{H})$ for $i,j=1,2\dots n$, then $\|A\|_{ber} \leq \left\|\left[\|A_{ij}\|_{ber}\right]\right\|$ and $\textbf{ber}(A) \leq w([a_{ij}]),$ where $a_{ii}=\textbf{ber}(A_{ii}),$ $a_{ij}=\big\||A_{ij}|+|A^*_{ji}|\big\|^{\frac{1}{2}}_{ber} \big\||A_{ji}|+|A^*_{ij}|\big\|^{\frac{1}{2}}_{ber}$ if $i<j$ and $a_{ij}=0$ if $i>j$. Further, we give some examples for the Berezin number and Berezin norm estimation of operator matrices on the Hardy-Hilbert space.