Berezin number and Berezin norm inequalities for operator matrices

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 $\Vert A{\Vert }_{ber}\le \Vert \left[\Vert A_{ij}{\Vert }_{ber}\right]\Vert$ and $\text{ber}(A)\le w([a_{ij}]),$ where $a_{ii}=\text{ber}(A_{ii}),$ $a_{ij}=\Vert |A_{ij}|+|A_{ji}^{\ast }|{\Vert }_{ber}^{\frac{1}{2}}\Vert |A_{ji}|+|A_{ij}^{\ast }|{\Vert }_{ber}^{\frac{1}{2}}$ 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.

References in corpus (2)

• Development of inequality and characterization of equality conditions for the numerical radius
• Convexity of the Berezin Range