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 $A=\left(\begin{array}{ccc}8& 1& 6\\ 3& 5& 7\\ 4& 9& 2\end{array}\right)$ which corresponds to a magic square has $\Vert A{\Vert }_{p,p}=\max \{\Vert A\xi {\Vert }_p:\xi \in {\ell }^p(n),\Vert \xi {\Vert }_p=1\}=15$ for any $p\in [1,\infty ]$. In this paper, we extend this result and we compute $\Vert A{\Vert }_{p,q}$ for $1\le q\le p\le \infty$.