Some generalizations of numerical radius on off-diagonal part of $2\times 2$ operator matrices

TL;DR

This paper extends inequalities related to the numerical radius for off-diagonal blocks of 2x2 operator matrices, providing new bounds involving operator norms and powers for such matrices.

Contribution

It introduces generalized inequalities for the numerical radius of off-diagonal 2x2 operator matrices, expanding existing bounds with new operator norm expressions.

Findings

Derived bounds for the r-th power of the numerical radius involving operator norms

Established inequalities for off-diagonal 2x2 operator matrices with specific operator combinations

Provided explicit bounds depending on parameters r and operator norms

Abstract

We generalize several inequalities involving powers of the numerical radius for off-diagonal part of $2\times2$ operator matrices of the form $T=\left[\begin{array}{cc} 0&B, C&0 \end{array}\right]$, where $B, C$ are two operators. In particular, if $T=\left[\begin{array}{cc} 0&B, C&0 \end{array}\right]$, then we get \begin{align*} {1\over 2^{{3\over2}(r-1)}}\max\{ \| \mu \|, \| \eta \| \} \leq w^{r}(T)\leq \frac{1}{2^{r+1}} \max\{ \| \mu \|, \| \eta \| \}, \end{align*} where $r\geq 2$ and $ \mu=|(C-B^{*})+i(C+B^{*})|^{r}+|(B^{*}-C)+i(C+B^{*})|^{r}$, $ \eta=|(B-C^{*})+i(B+C^{*})|^{r}+|(C^{*}-B)+i(B+C^{*})|^{r}$.