von Neumann’s inequality for row contractive matrix tuples

Michael Hartz, Stefan Richter and I recently uploaded our paper von Neumann’s inequality for row contractive matrix tuples to the arxiv.

The main result is the following.

We prove that for all $latex d,nin mathbb{N}$, there exists a constant $latex C_{d,n}$ such that for every row contraction $latex T$ consisting of $latex d$ commuting $latex n times n$ matrices and every polynomial $latex p$, the following inequality holds:

$latex  |p(T)| le C_{d,n} sup_{z in mathbb{B}_d} |p(z)|$ .

We then apply this result and the considerations involved in the proof to several open problems from the literature. I won’t go into that because I think that the abstract and introduction do a good job of explaining what we do in the paper. In this post I will write about how this collaboration with Michael and Stefan started, and give some heuristic explanation why our result is not trivial.