Index of Hadamard multiplication by positive matrices II

Given a definite nonnegative matrix $A \in M_n (C)$, we study the minimal index of A: $I(A) = \max \{\lambda \ge 0 : A\circ B \ge \lambda B$ for all $0\le B\}$, where $A\circ B$ denotes the Hadamard product $(A\circ B)_{ij} = A_{ij} B_{ij}$. For any unitary invariant norm N in $M_n(C)$, we consider the N-index of A: $I(N,A) = \min\{N(A\circ B) : B\ge 0$ and $N(B) = 1 \}$. If A has nonnegative entries, then $I(A) = I(\| \cdot \|_{sp}, A)$ if and only if there exists a vector u with nonnegative entries such that $Au = (1, >..., 1)^T$. We also show that $I(\| \cdot \|_{2}, A)= I(\| \cdot \|_{sp}, {\bar A}\circ A)^{1/2}$. 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 + S^{-1} T S^{-1} \| \ge M(S) \|T\| $ for all $0 \le T$.