Norms of structured random matrices

For $m,n\in N$ , let $X={\left(X_{\mathrm{ij}}\right)}_{i\le m,j\le n}$ be a random matrix, $A={\left(a_{\mathrm{ij}}\right)}_{i\le m,j\le n}$ a real deterministic matrix, and $X_A={\left(a_{\mathrm{ij}}{X}_{\mathrm{ij}}\right)}_{i\le m,j\le n}$ the corresponding structured random matrix. We study the expected operator norm of $X_A$ considered as a random operator between ${\ell }_p^n$ and ${\ell }_q^m$ for $1\le p,q\le \infty$ . We prove optimal bounds up to logarithmic terms when the underlying random matrix X has i.i.d. Gaussian entries, independent mean-zero bounded entries, or independent mean-zero ${\psi }_r$ ( $r\in (0,2]$ ) entries. In certain cases, we determine the precise order of the expected norm up to constants. Our results are expressed through a sum of operator norms of Hadamard products $A\circ A$ and ${(A\circ A)}^T$ .