Inequalities for operator space numerical radius of $2\times 2$ block matrices

TL;DR

This paper investigates inequalities relating the operator space numerical radius and norm for 2x2 block matrices, providing bounds and relationships in the context of operator space theory.

Contribution

It establishes new inequalities connecting the operator space numerical radius with the maximal numerical radius norm for 2x2 matrices, advancing understanding in operator space analysis.

Findings

Derived bounds for the numerical radius of 2x2 block matrices.

Connected the numerical radius with the maximal numerical radius norm.

Provided inequalities involving off-diagonal parts of matrices.

Abstract

In this paper, we study the relationship between operator space norm and operator space numerical radius on the matrix space $\mathcal{M}_n(X)$, when $X$ is a numerical radius operator space. Moreover, we establish several inequalities for operator space numerical radius and the maximal numerical radius norm of $2\times 2$ operator matrices and their off-diagonal parts. One of our main results states that if $(X, (O_n))$ is an operator space, then \begin{align*} \frac12\max\big(W_{\max}(x_1+x_2)&, W_{\max}(x_1-x_2) \big)\\ &\le W_{\max}\Big(\begin{bmatrix} 0 & x_1 \\ x_2 & 0 \end{bmatrix}\Big)\\ &\hspace{1.5cm}\le \frac12\left(W_{\max}(x_1+x_2)+ W_{\max}(x_1-x_2) \right) \end{align*} for all $x_1, x_2\in \mathcal{M}_n(X)$.