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.

[1] T. Ando, On the structure of operators with numerical radius one, Acta Sci. Math., (Szeged) 34 (1973), 11–15. [2] T. Ando and K. Okubo, Induced norms of the Schur multiplier operator, Linear Algebra Appl., 147 (1991), 181–199. [3] J.R. Angelos, C.C. Cowen and S.K. Narayan, Triangular truncation and finding the norm of a Hadamard multiplier, Linear Algebra Appl., 170 (1992), 181–199. [4] K.M.R. Audenaert, Schur multiplier norms for Loewner matrices, Linear Algebra Appl., 439 (2013), 2598– 2608. [5] R. Bhatia, Positive Definite Matrices, Princeton University Press, Princeton, 2007. [6] P. Chaisuriya, A C∗-algebra on Schur algebras, Bull. Malays. Math. Sci. Soc., 34(2) (2011), 189-200. [7] P. Chaisuriya and S.-C. Ong, On Schur Multiplier Norm and Unitaries, Southeast Asian Bull. Math., 26 (2003), no. 6, 889–898. [8] C.K. Fong, H. Radjavi and P. Rosenthal, norms for matrices and operators, J. Operator Theory, 18 (1987), 99–113. [9] I.C. Gohberg and M.G. Krein, Theory and applications of Volterra operators in Hilbert space, English translation, AMS, Providence, 1970. [10] R.A. Horn and C.R. Johnson, Matrix Analysis, Second ed. Cambridge university press, 2012. [11] A. Katavolos and V. Paulsen, On the ranges of bimodule projections, Canad. Math. Bull., 48(1)(2005), 91–111. [12] L. Livshits, Generalized Schur Products for Matrices with Operator Entries, ProQuest LLC, Ann Arbor, MI, 1991. [13] L. Livschits, A note on 0-1 Schur multipliers, Linear Algebra Appl. 22 (1995), 15–22. [14] R. Mathias, An arithmetic-geometric-harmonic mean inequality involving Hadamard products, Linear Algebra Appl. 184 (1993), 71–78. [15] S.-C. Ong, On the Schur multiplier of matrices, Linear Algebra Appl. 56 (1984), 45–55.