von Neumann's inequality for row contractive matrix tuples
Michael Hartz, Stefan Richter, Orr Shalit
TL;DR
The paper extends von Neumann’s inequality to commuting row contractions built from $d$ commuting $n\times n$ matrices, establishing a dimension-free constant $C_n$ so that $\|p(T)\| \le C_n \|p\|_\infty$ for all polynomials $p$. It develops a Schur-algorithm–inspired approach via Gleason’s problem, yielding explicit finite bounds and revealing sharp behavior in low dimensions (e.g., $C_{d,2}=1$) while showing growth for larger $n$ when $d\ge2$. The work uses these operator-theoretic tools to resolve several open problems: Gleason’s problem is not contractively solvable in $H^\infty(\mathbb B_d)$ for $d\ge2$, multiplier algebras on weighted Dirichlet spaces are not topologically subhomogeneous for $0<a<d$, and there exist uniformly bounded nc holomorphic functions on the free commutative ball that are levelwise but not globally uniformly continuous. Collectively, the results illuminate connections between multivariable operator theory, RKHS on the ball, and nc function theory, with implications for finite-dimensional representations and nc continuity phenomena.
Abstract
We prove that for all $n\in \mathbb{N}$, there exists a constant $C_{n}$ such that for all $d \in \mathbb{N}$, for every row contraction $T$ consisting of $d$ commuting $n \times n$ matrices and every polynomial $p$, the following inequality holds: \[ \|p(T)\| \le C_{n} \sup_{z \in \mathbb{B}_d} |p(z)| . \] 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 $H^\infty(\mathbb{B}_d)$ for $d \ge 2$. Second, we prove that the multiplier algebra $\operatorname{Mult}(\mathcal{D}_a(\mathbb{B}_d))$ of the weighted Dirichlet space $\mathcal{D}_a(\mathbb{B}_d)$ on the ball is not topologically subhomogeneous when $d \ge 2$ and $a \in (0,d)$. In fact, we determine all the bounded finite dimensional representations of the norm closed subalgebra $A(\mathcal{D}_a(\mathbb{B}_d))$ of $\operatorname{Mult}(\mathcal{D}_a(\mathbb{B}_d))$ generated by polynomials. Lastly, we also show that there exists a uniformly bounded nc holomorphic function on the free commutative ball $\mathfrak{C}\mathfrak{B}_d$ that is levelwise uniformly continuous but not globally uniformly continuous.