Noncommutative Analysis

Tag: von Neumann inequality

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 d,n\in \mathbb{N}, there exists a constant C_{d,n} such that 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_{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.

Read the rest of this entry »

Spectral sets and distinguished varieties in the symmetrized bidisc

In this post I will write about a new paper, “Spectral sets and distinguished varieties in the symmetrized bidisc“, that Sourav Pal and I posted on the arxiv, and give the background to understand what we do in that paper.

Read the rest of this entry »