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 ο B ⪰ λB for all B ⪰ 0} and, for each norm N, the N-index I_N(A) = min{N(A ο B): B ⪰0 and N(B) = 1}, where A ο B = [aij bij] is the Hadamard or Schur product of A =[aij] and B = [bij] 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 ∥ST S + S^−1T S^−1∥ M(S)∥T∥ for all T⪰ 0.