Index of Hadamard multiplication by positive matrices II

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 IN(A)=min{N(A∘B):B⪰0 and N(B)=1}, where A∘B=[aijbij] 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 ∥STS+S−1TS−1∥⩾M(S)∥T∥ for all T⪰0.