Improvements of Berezin number inequalities

TL;DR

This paper extends several inequalities related to the Berezin number, particularly involving products of operators, providing generalized bounds and relations for positive operators and certain operator functions.

Contribution

It introduces new generalized inequalities for the Berezin number involving operator products and functions, broadening the scope of existing bounds in operator theory.

Findings

Derived new bounds for Berezin numbers of operator products

Established inequalities involving positive operators and operator functions

Generalized existing Berezin number inequalities to broader classes

Abstract

In this paper, we generalize several Berezin number inequalities involving product of operators. For instance, we show that if $A, B$ are positive operators and $X$ is any operator, then \begin{align*} \textbf{ber}^{r}(H_{\alpha}(A,B))&\leq\frac{\|X\|^{r}}{2}\textbf{ber}(A^{r}+B^{r})&\leq\frac{\|X\|^{r}}{2}\textbf{ber}(\alpha A^{r}+(1-\alpha)B^{r})+\textbf{ber}((1-\alpha)A^{r}+\alpha B^{r}), \end{align*} where $H_{\alpha}(A,B)=\frac{A^\alpha XB^{1-\alpha}+A^{1-\alpha} XB^{\alpha}}{2}$, $0\leq\alpha\leq1$ and $r\geq2$.