Let $A=(a_{i,j}):\ell_\infty\to\ell_1$ and $B=(b_{i,j}):\ell_1\to\ell_\infty$ be linear operators. We can define their operator/matrix norms as follows: $$ \|A\|_{\infty,1} =\sup\left\{\sum_{i=1}^\infty\left|\sum_{j=1}^\infty a_{i,j}x_j \right|:\sup|x_j|\leq 1\right\} $$ and $$ \|B\|_{1,\infty} =\sup\left\{\sup\left|\sum_{j=1}^\infty b_{i,j}x_j\right|:\sum_{j=1}^\infty|x_j|\leq 1\right\} $$ Are there any "nicer" formulas for these norms which can be written only in terms of $(a_{i,j})$ and $(b_{i,j})$, without reference to arbitrary vectors $(x_j)$? I vaguely remember that there are such formulas, but I can't seem to find them today.
Thanks!
Hint: For $|x_j|\leq 1$, $$\left|\sum_{j=1}^\infty a_{i,j}x_j \right|\le\sum_{j=1}^\infty|a_{i,j}|$$ with equality when $x_j=\mathrm{sgn}(a_{i,j})$.