Index of Hadamard Multiplication by Positive Matrices Ii

Given a definite nonnegative matrix A ∈ Mn(C), we study the minimal index of A : I(A) = max{λ ≥ 0 : A ◦ B ≥ λB for all 0 ≤ B}, where A ◦ B denotes the Hadamard product (A ◦ B)ij = AijBij . For any unitary invariant norm N in Mn(C), we consider the Nindex of A: I(N,A) = min{N(A ◦ B) : B ≥ 0 and N(B) = 1} If A has nonnegative entries, then I(A) = I(‖ · ‖sp, A) if and only if there exists a vector u with nonnegative entries such that Au = (1, . . . , 1)T . We also show that I(‖ · ‖2, A) = I(‖ · ‖sp, Ā◦A)1/2. We give formulae for I(N,A), for an arbitrary unitary invariant norm N , when A is a diagonal matrix or a rank 1 matrix. As an application we find, for a bounded invertible selfadjoint operator S on a Hilbert space, the best constant M(S) such that ‖STS + STS‖ ≥ M(S)‖T‖ for all 0 ≤ T .