Inequalities of the <i>ℓ</i> ^<i>p</i> -operator norm for block matrices

Abstract We develop several $\ell ^p$ -operator norm inequalities for $k\times k$ block matrices defined on the $\ell ^p$ -sum of Banach spaces. Using these inequalities, we obtain p -numerical radius and spectral radius bounds for $k\times k$ block matrices. We deduce a p -numerical radius bound for the Kronecker product $A\otimes B$ , where $A\in {M}_k(\mathbb {C})$ is a $k\times k$ complex matrix and $B\in \mathcal {L}(\mathbb {H})$ is a bounded linear operator on a complex Hilbert space $\mathbb {H}$ . This improves and extends Holbrook’s bound $w(A\otimes B)\leq w(A)\|B\|.$ If $\|A\|_{\ell ^p}$ and $w_p(A)$ denote the $\ell ^p$ -operator norm and p -numerical radius of $A\in {M}_k(\mathbb {C})$ , respectively, then it is shown that $$ \begin{align*} \frac{1}{2}\|A\|_p+\mu_p(A) \leq w_p(A), \end{align*} $$ where $\mu _p(A)$ is a positive real number that involves the $\ell ^p$ -operator norms of the Cartesian decomposition of A . In addition, a complete characterization of the equality case $\frac {1}{2}\|A\|_p= w_p(A)$ is given.