Schur multiplier norm of product of matrices

For a ∈ mn, the schur multiplier of a is defined as s a(x) =a ◦ x for all x ∈ mn and the spectral norm of s a can be stateas ∥s a∥ = supx,0 ∥a ∥x ◦x ∥ ∥. the other norm on s a can be definedas ∥s a∥ω = supx,0 ω(ω s( ax (x ) )) = supx,0 ωω (a (x ◦x ) ), where ω(a) standsfor the numerical radius of a. in this paper, we focus on therelation between the norm of schur multiplier of product of matrices and the product of norm of those matrices. this relation isproved for schur product and geometric product and some applications are given. also we show that there is no such relationfor operator product of matrices. furthermore, for positive definite matrices a and b with ∥s a∥ω ⩽ 1 and ∥s b∥ω ⩽ 1, we showthat a♯b = n(i − z)1/2c(i + z)1/2, for some contraction c andhermitian contraction z.