Schur multiplier norm of product of matrices
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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.
کلید واژگان
Schur multiplierSchur product
Geometric product
Positive semidefinite matrix
Numerical radius
شماره نشریه
1تاریخ نشر
2015-09-011394-06-10
ناشر
Vali-e-Asr university of Rafsanjanسازمان پدید آورنده
Shahid Bahonar university of KermanShahid Bahonar university of Kerman
شاپا
2383-19362476-3926
