The operator $(p, q)$-norm of some matrices

Abstract

We compute the operator $(p,q)$-norm of some $n\times n$ complex matrices, which can be seen as bounded linear operators from the $n$ dimensional Banach space $\ell^p(n)$ to $\ell^q(n)$. We have shown that a special matrix $A=\begin{pmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{pmatrix}$ which corresponds to a magic square has $\|A\|_{p,p} = \max \{\|A\xi\|_p : \xi\in\ell^p(n), \|\xi\|_p=1\} =15$ for any $p\in [1,\infty]$. In this paper, we extend this result and we compute $\|A\|_{p,q}$ for $1\le q \le p \le \infty$.