On Von Neumann’s Inequality for Matrices of Complex Polynomials

Abstract

We prove that every matrix F ∈ M k (P n ) is associated with the smallest positive integer d ( F ) ≠ 1 such that d ( F ) ‖ F ‖ ∞ is always bigger than the sum of the operator norms of the Fourier coefficients of F . We establish some inequalities for matrices of complex polynomials. In application, we show that von Neumann’s inequality hold s up to the constant 2 n for matrices of the algebra M k (P n ).