Schur multiplier norm of product of matrices

Document Type : Research Paper

Authors

Shahid Bahonar university of Kerman

Abstract

For A ∈ M n, the Schur multiplier of A is defined as S A(X) = A ◦ X for all X ∈ M n and the spectral norm of S A can be state as ∥S A∥ = supX,0 ∥A ∥X ◦X ∥ ∥. The other norm on S A can be defined as ∥S A∥ω = supX,0 ω(ω S( AX (X ) )) = supX,0 ωω (A (X ◦X ) ), where ω(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 matrices and the product of norm of those matrices. This relation is proved for Schur product and geometric product and some applications are given. Also we show that there is no such relation for operator product of matrices. Furthermore, for positive definite matrices A and B with ∥S A∥ω ⩽ 1 and ∥S B∥ω ⩽ 1, we show that A♯B = n(I − Z)1/2C(I + Z)1/2, for some contraction C and Hermitian contraction Z.

Keywords

References

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Wavelet and Linear Algebra
Volume 2, Issue 1 - Serial Number 1
September 2015
Pages 49-54

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