# Upper bounds for numerical radius inequalities involving off-diagonal operator matrices

## TL;DR

This paper derives new upper bounds for the numerical radius of off-diagonal 2x2 operator matrices, involving functions of operators and generalized Euclidean radii, advancing the understanding of operator inequalities.

## Contribution

It introduces novel upper bounds for the numerical radius of off-diagonal operator matrices using functions f and g, and explores inequalities for generalized Euclidean operator radius.

## Key findings

- Established bounds for numerical radius involving off-diagonal blocks.
- Derived inequalities using functions satisfying f(t)g(t)=t.
- Extended results to generalized Euclidean operator radius.

## Abstract

In this paper, we establish some upper bounds for numerical radius inequalities including of $2\times 2$ operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if $T=\left[\begin{array}{cc} 0&X, Y&0 \end{array}\right]$, then \begin{align*} \omega^{r}(T)\leq 2^{r-2}\left\|f^{2r}(|X|)+g^{2r}(|Y^*|)\right\|^\frac{1}{2}\left\|f^{2r}(|Y|)+g^{2r}(|X^*|)\right\|^\frac{1}{2} \end{align*} and \begin{align*} \omega^{r}(T)\leq 2^{r-2}\left\|f^{2r}(|X|)+f^{2r}(|Y^*|)\right\|^\frac{1}{2}\left\|g^{2r}(|Y|)+g^{2r}(|X^*|)\right\|^\frac{1}{2}, \end{align*} where $X, Y$ are bounded linear operators on a Hilbert space ${\mathscr H}$, $r\geq 1$ and $f$, $g$ are nonnegative continuous functions on $[0, \infty)$ satisfying the relation $f(t)g(t)=t\,(t\in[0, \infty))$. Moreover, we present some inequalities involving the generalized Euclidean operator radius of operators $T_{1},\cdots,T_{n}$.