Index of Hadamard multiplication by positive matrices II

Abstract

For each n×n positive semidefinite matrix A we define the minimal index I(A)= max{λ⩾0:A∘B⪰λB for all B⪰0} and, for each norm N, the N-index I N(A)= min{N(A∘B):B⪰0 and N(B)=1}, where A∘B=[a ijb ij] is the Hadamard or Schur product of A=[a ij] and B=[b ij] and B⪰0 means that B is a positive semidefinite matrix. A comparison between these indexes is done, for different choices of the norm N. As an application we find, for each bounded invertible selfadjoint operator S on a Hilbert space, the best constant M(S) such that ∥STS+S −1TS −1∥⩾M(S)∥T∥ for all T⪰0.