Index of Hadamard multiplication by positive matrices II

Abstract

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 $I(A)=I(\Vert \cdot {\Vert }_{sp},A)$ if and only if there exists a vector u with nonnegative entries such that $Au=(1,>...,1{)}^T$. We also show that $I(\Vert \cdot {\Vert }_2,A)=I(\Vert \cdot {\Vert }_{sp},\bar{A}\circ A{)}^{1/2}$. 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 $\Vert STS+S^{-1}T{S}^{-1}\Vert \ge M(S)\Vert T\Vert$ for all $0\le T$.

19 pages, Latex