Inequalities for operator space numerical radius of $2\times 2$ block 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 $x_1,x_2\in {\mathcal{M}}_n(X)$.

to appear in J. Math. Phys. (JMP), 18 pages

References in corpus (2)

• Cartesian decomposition and Numerical radius inequalities
• Restricted numerical shadow and geometry of quantum entanglement

Cited by in corpus (3)

• Seminorm and numerical radius inequalities of operators in semi-Hilbertian spaces
• Characterization of numerical radius parallelism in $C^{\ast }$-algebras
• Some upper bounds for the $\mathbb{A}$-numerical radius of $2\times 2$ block matrices