Some spectral norm inequalities on Hadamard products of nonnegative matrices

Let A and B be nonnegative square matrices of the same order. Denote by ‖⋅‖ and ρ(⋅) the spectral norm and the spectral radius respectively. We prove the following inequalities:‖A∘B‖≤‖A∘A‖12‖B∘B‖12;‖A∘B‖≤ρ12(ATB∘BTA)≤ρ12(ATB∘ATB)≤ρ(ATB), where ∘ denotes the Hadamard product. This interpolates the inequality‖A∘B‖≤ρ(ATB) due to Huang. Some spectral norm inequalities for an arbitrarily finite number of nonnegative square matrices are also obtained, which refine some other results of Huang.