Empirical bounds for commuting dilations of free unitaries and the universal commuting dilation constant
This work addresses a theoretical problem in operator theory, providing incremental insights into dilation constants for commuting operators.
The paper tackles the problem of determining the commuting dilation constant for pairs of contractions, with numerical experiments suggesting it converges to √2 for random unitaries as N→∞, and proves that for arbitrary pairs of contractions, the constant is strictly less than 2 under this assumption.
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.