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.