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

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 ${\mathrm{\ell }}^p(n)$ to ${\mathrm{\ell }}^q(n)$. 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 $\mathrm{\parallel }A{\mathrm{\parallel }}_{p,p}=\max \{\mathrm{\parallel }A\xi {\mathrm{\parallel }}_p:\xi \in {\mathrm{\ell }}^p(n),\mathrm{\parallel }\xi {\mathrm{\parallel }}_p=1\}=15$ for any $p\in [1,\mathrm{\infty }]$. In this paper, we extend this result and we compute $\mathrm{\parallel }A{\mathrm{\parallel }}_{p,q}$ for $1\le q\le p\le \mathrm{\infty }$.