We prove that every
matrix F∈Mk (Pn) is associated withthe 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 holds up to the constant 2n for matrices of the algebraMk (Pn).
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