Schur test ($\Vert A{\Vert }_{{\ell }^2\to {\ell }^2}\le \sqrt{RC}$)

For a matrix with nonnegative entries, row sums at most $R$ and column sums at most $C$ imply an ${\ell }^2$ operator norm at most $\sqrt{RC}$. Indeed the Cauchy-Schwarz inequality gives $|{\sum }_j{A}_{ij}{x}_j{|}^2\le ({\sum }_j{A}_{ij}){\sum }_j{A}_{ij}|x_j{|}^2$. Summing over $i$ bounds the result by $RC{\sum }_j|x_j{|}^2$. Integral kernels and positive weight functions give analogous forms. The symmetric case has equal row and column bounds.