Index of Hadamard multiplication by positive matrices II

CORACH, GUSTAVO; STOJANOFF, DEMETRIO

LINEAR ALGEBRA AND ITS APPLICATIONS

ELSEVIER SCIENCE INC

Año: 2001 vol. 332-334 p. 503 - 517

0024-3795

For each n×n positive semidefinite matrix A we define the minimal index I(A)=max{λ≥0:ABλB for all B0} and, for each norm N, the N-index IN(A)=min{N(AB):B0 and N(B)=1}, where AB=[aijbij] is the Hadamard or Schur product of A=[aij] and B=[bij] and B0 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 T0.