Index of Hadamard multiplication by positive matrices II

For each n x n positive semidefinite matrix A we define the minimal index I (A) = max{lambda greater than or equal to 0: A o B greater than or equal to lambdaB for all B greater than or equal to 0} and, for each norm N, the N- index I-N (A) = min{N(A o B) : B greater than or equal to 0 and N(B) = 1}, where A o B = [a(ij)b(ij)] is the Hadamard or Schur product of A = [a(ij)] and B = [b(ii)] and B greater than or equal to 0 means that B is a positive semidefinite matrix. A comparison between these indexes is done, for diffe rent choices of the norm N. As an application we find, for each bounded inv ertible selfadjoint operator S on a Hilbert space, the best constant M(S) s uch that //STS + S-1 TS-1// greater than or equal to M(S)//T// for all T gr eater than or equal to 0. (C) 2001 Published by Elsevier Science Inc.