Empirical bounds for commuting dilations of free unitaries and the universal commuting dilation constant
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| Format: | Preprint |
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2025
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| _version_ | 1866914397165715456 |
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| author | Gerhold, Malte Scherer, Marcel Shalit, Orr |
| author_facet | Gerhold, Malte Scherer, Marcel Shalit, Orr |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_12540 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Empirical bounds for commuting dilations of free unitaries and the universal commuting dilation constant Gerhold, Malte Scherer, Marcel Shalit, Orr Functional Analysis Numerical Analysis Operator Algebras 47A20 (Primary) 46L07, 46L89, 60B20, 90C22 (Secondary) 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. |
| title | Empirical bounds for commuting dilations of free unitaries and the universal commuting dilation constant |
| topic | Functional Analysis Numerical Analysis Operator Algebras 47A20 (Primary) 46L07, 46L89, 60B20, 90C22 (Secondary) |
| url | https://arxiv.org/abs/2510.12540 |