Abstract
For m, n ∈ N , let X = (Xij)i≤m,j≤n be a random matrix, A = (aij)i≤m,j≤n a real deterministic matrix, and XA = (aijXij)i≤m,j≤n the corresponding structured random matrix. We study the expected operator norm of XA considered as a random operator between ℓpn and ℓqm for 1 ≤ p, q ≤ ∞ . 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 ψr ( r ∈ (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 ∘ A and (A∘A)T .