On the spectral radius and the spectral norm of Hadamard products of nonnegative matrices

We prove the spectral radius inequality ρ(A1∘A2∘⋯∘Ak)⩽ρ(A1A2⋯Ak) for nonnegative matrices using the ideas of Horn and Zhang. We obtain the inequality ‖A∘B‖⩽ρ(ATB) for nonnegative matrices, which improves Schur’s classical inequality ‖A∘B‖⩽‖A‖‖B‖, where ‖·‖ denotes the spectral norm. We also give counterexamples to two conjectures about the Hadamard product.