von Neumann’s inequality for row contractive matrix tuples

Abstract

We prove that for all n∈N, there exists a constant Cn such that for all d∈N, for every row contraction T consisting of d commuting n×n matrices and every polynomial p, the following inequality holds:

∥p(T)∥≤Cnsupz∈Bd|p(z)|. We apply this result and the considerations involved in the proof to several open problems from the pertinent literature. First, we show that Gleason’s problem cannot be solved contractively in H∞(Bd) for d≥2. Second, we prove that the multiplier algebra Mult(Da(Bd)) of the weighted Dirichlet space Da(Bd) on the ball is not topologically subhomogeneous when d≥2 and a∈(0,d). In fact, we determine all the bounded finite dimensional representations of the norm closed subalgebra A(Da(Bd)) of Mult(Da(Bd)) generated by polynomials. Lastly, we also show that there exists a uniformly bounded nc holomorphic function on the free commutative ball CBd that is levelwise uniformly continuous but not globally uniformly continuous.