Schur multiplier norm of product of matrices
Maryam Khosravi, Alemeh Sheikhhosseini · 2015
For A 2 Mn, the Schur multiplier of A is defined as S A(X) = AX for all X 2 Mn and the spectral norm of S A can be state as ∥S A∥ = sup X,0 ∥AX∥ ∥X∥ . The other norm on S A can be defined as ∥S A∥! = sup X,0 ! (S A(X)) ! (X) = sup X,0 ! (AX) ! (X) , where ! (A) stands for the numerical radius of A. In this paper, we focus on the relation between the norm of Schur multiplier of product of ma- trices and the product of norm of those matrices. This relation is proved for Schur product and geometric product and some ap- plications are given. Also we show that there is no such relation for operator product of matrices. Furthermore, for positive defi- nite matrices A and B with ∥S A∥! ⩽ 1 and ∥S B∥! ⩽ 1, we show that A♯B = n(I Z) 1= 2 C(I + Z) 1= 2 ; for some contraction C and Hermitian contraction Z: c ⃝ (2015) Wavelets and Linear Algebra