von Neumann’s inequality for row contractive matrix tuples

Michael Hartz, Stefan Richter, Orr Moshe Shalit

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish
Pages (from-to)3877-3894
Number of pages18
JournalMathematische Zeitschrift
Issue number4
StatePublished - 1 Aug 2022
Externally publishedYes


  • Gleason’s problem
  • Noncommutative function theory
  • Von Neumann type inequality

ASJC Scopus subject areas

  • Mathematics (all)


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