Empirical bounds for commuting dilations of free unitaries and the universal commuting dilation constant

Malte Gerhold; Marcel Scherer; Orr Shalit

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arXiv · 2510.12540

Empirical bounds for commuting dilations of free unitaries and the universal commuting dilation constant

Malte Gerhold·Marcel Scherer·Orr Shalit

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Abstract

For a tuple $T$ of Hilbert space operators, the 'commuting dilation constant' is the smallest number $c$ such that the operators of $T$ are a simultaneous compression of commuting normal operators of norm at most $c$. We present numerical experiments giving a strong indication that the commuting dilation constant of a pair of independent random $N{\times}N$ unitary matrices converges to $\sqrt2$ as $N \to \infty$ almost surely. Under the assumption that this is the case, we prove that the commuting dilation constant of an arbitrary pair of contractions is strictly smaller than $2$. Our experiments are based on a simple algorithm that we introduce for the purpose of computing dilation constants between tuples of matrices.

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BibTeX RIS

Malte Gerhold, Marcel Scherer, Orr Shalit. 2025-10-14. Empirical bounds for commuting dilations of free unitaries and the universal commuting dilation constant. https://arxiv.org/abs/2510.12540

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