Commuting Dilation Constant for Operator Tuples

• The topic defines the commuting dilation constant as the minimal scaling that allows any tuple of contractions on a Hilbert space to dilate to a commuting normal tuple.
• It employs numerical simulation and semidefinite programming to approximate the constant, notably showing that for operator pairs the value is empirically near √2.
• The insights refine multivariable von Neumann inequalities and inform applications in operator theory, control systems, and quantum information.

The commuting dilation constant is a central numerical invariant in multivariable operator theory, quantifying the minimal scale by which operator tuples must be amplified to admit a commuting normal (or unitary) dilation. For a tuple $T = (T_1, ..., T_d)$ of Hilbert space operators, the commuting dilation constant is the smallest $c \geq 1$ such that $T$ can be realized as a simultaneous compression of a $d$-tuple of commuting normal operators of norm at most $c$. This notion captures the gap between general (possibly noncommuting or nonnormal) dynamics and those generated by commuting normal operators, and it governs sharp constants in multivariate von Neumann–type inequalities, matrix convexity, and dilation-theoretic relaxations. Recent work (Gerhold et al., 14 Oct 2025) combines algorithmic computation, probabilistic spectral analysis, and new theoretical estimates to refine the known bounds for the universal commuting dilation constant in low dimensions, providing strong evidence for its exact value in the case of pairs of operators.

1. Formal Definition and Universal Constant

Given a tuple $T = (T_1,...,T_d)$ of Hilbert space operators, the (universal) commuting dilation constant $C_d$ is defined by

$$C_d = \inf\{c \geq 1 : \text{every } d\text{-tuple of contractions dilates to a commuting normal tuple } N \text{ with } \|N_j\| \leq c \text{ for all } j\}.$$

Equivalently, $C_d$ is the optimal scaling for which, for any $T$ in this class, there exists a commuting normal tuple $N = (N_1, ..., N_d)$ and an isometry $V$ such that

$T_j = V^* N_j V, \qquad \|N_j\| \leq C_d,$

for all $1 \leq j \leq d$. This is often phrased in terms of matrix-valued dilation constants between tuples of unitaries: $C_d = c(u_\mathrm{u}, u_0)$, where $u_\mathrm{u}$ denotes the universal (free) $d$-tuple of unitaries and $u_0$ the universal commuting $d$-tuple (generating $C(\mathbb T^d)$).

2. Approaches to Computing Dilation Constants

The precise computation of the commuting dilation constant is intractable via direct functional calculus in infinite dimensions. The methodology in (Gerhold et al., 14 Oct 2025) is based on:

• Large-scale numerical simulation: for $d=2$, independent pairs of Haar-distributed random $N \times N$ unitary matrices $U = (U_1,U_2)$ are sampled for increasing $N$, and the dilation constant $c(U,u_0)$ is estimated.
• Finite-dimensional convex optimization: the computation reduces to a semidefinite program for the maximal $r$ such that there exists a unital completely positive (UCP) map $\Phi$ satisfying

$\Phi(N_i) = r U_i, \qquad (i=1,2),$

where $N = (N_1,N_2)$ is a fixed pair of commuting normal matrices with spectra discretizing the torus (e.g., through $k$-gons).

• Theoretical estimates: rigorous bounds are derived by exploiting the convergence in matrix ranges of large random unitary tuples to those of free Haar unitary tuples $u_f$, and leveraging known dilation constants between universal and free Haar tuples.

The algorithmic procedure is summarized by the following convex program:

Step	Description
Fix $U = (U_1, U_2)$	Given $N \times N$ Haar unitaries
Fix $N = (N_1, N_2)$	Commuting normal matrices with spectrum in discretization $V_k$
Seek maximizer $r$	Subject to a UCP map $\Phi$ with $\Phi(N_i) = r U_i$
Implementation	SDP over $k^2$ variables $C_j \geq 0$ satisfying:
	(i) $\sum_j C_j = I_N$
	(ii) $\sum_j (N_i)_{jj} C_j = r U_i$ ($i=1,2$)
Output	$c(U,N) = r^{-1}$ (estimates $c(U, u_0)$)

The optimization is efficiently solvable for moderate $N$, $k$ using standard convex solvers.

3. Empirical and Theoretical Results for $d=2$

The computed values for large $N$ consistently and robustly cluster near $\sqrt{2}$, regardless of numeric realization. Combined with known convergence properties of matrix ranges ((Gerhold et al., 14 Oct 2025), Theorem 3.1), this provides strong evidence that

$c(u_f, u_0) = \sqrt{2}$

where $u_f$ denotes a pair of free Haar unitaries. The universal constant,

$$C_2 = c(u_\mathrm{u}, u_0) \leq c(u_\mathrm{u}, u_f) \cdot c(u_f,u_0) = \frac{2}{\sqrt{3}} \cdot \sqrt{2} = 2 \sqrt{\frac{2}{3}} < 2,$$

exploits $c(u_\mathrm{u},u_f)=2/\sqrt{3}$ and the triangle inequality from the theory of matrix ranges.

A finite-dimensional bound (Appendix, (Gerhold et al., 14 Oct 2025)) confirms

$$C_2(n) \leq \sqrt{2 + 2 \sin\left(\frac{\pi}{2}(1 - 1/(2n))\right)} < 2,$$

for every $n \in \mathbb N$, reinforcing that the dilation constant remains strictly below the trivial upper bound of 2 as $n\to\infty$.

4. Relation to Previous Bounds and Operator-Theoretic Significance

Prior results provided only sandwich bounds. It was previously known that for $d$-tuples of contractions,

$\sqrt{d} \leq C_d \leq \sqrt{2d}$

with the upper bound from the Passer estimate and the lower from explicit non-dilatable constructions.

The empirical finding that $C_2 = \sqrt{2}$ in the free limit suggests a substantial gap between the trivial upper bound ($2$) and the actual best constant, thereby refining our understanding of the minimal scaling needed for commuting normal dilations of general pairs of contractions.

This has broad implications:

• The von Neumann-type inequality for pairs of contractions is sharp with constant $\sqrt{2}$, not $2$.
• Any operator-theoretic procedure (e.g., control system design, noncommutative convexity, free spectrahedra relaxations) that reduces noncommuting problems to commuting ones via dilation, can employ tighter constants for accuracy and efficiency.

5. Open Questions and Future Directions

Several key issues remain open:

• Fully rigorous proof of almost sure convergence of the dilation constant $c(U^{(N)}, u_0) \to \sqrt{2}$ for Haar random unitary pairs.
• Explicit computation or characterization of $C_d$ for $d > 2$: empirical and theoretical approaches must be refined for higher dimensions.
• Understanding the precise mechanisms connecting the combinatorics of matrix ranges, matrix convex sets, and dilation constants, particularly as $d$ increases.
• Potential applications to quantum information, where optimal dilations control quantum channel simulation and measurement compatibility.

The work also motivates further numerical, probabilistic, and geometric investigations into matrix ranges, joint spectra of random operator tuples, and the behavior of unital completely positive maps in high dimensions.

6. Key Formulas

• Universal commuting dilation constant

$C_d = c(u_\mathrm{u}, u_0)$

where $u_\mathrm{u}$ is the free $d$-tuple of unitaries, $u_0$ is the universal commuting $d$-tuple.

• Empirical limiting value for $d=2$

$C_2 = \sqrt{2}$

supported by numerical experiments.

• Finite-dimensional upper bound

$$C_2(n) \leq \sqrt{2 + 2 \sin\left(\frac{\pi}{2}(1 - 1/(2n))\right)} < 2$$

for all $n$.

• Triangle inequality for dilation constants

$$c(u_\mathrm{u}, u_0) \leq c(u_\mathrm{u}, u_f) \cdot c(u_f, u_0)$$

where $u_f$ is the free Haar unitary tuple.

• Semidefinite program for computing $c(U,N)$

$$\begin{cases} \text{maximize } r \ \text{subject to } \sum_{j=1}^{k^2} C_j = I, \quad C_j \geq 0, \ \sum_{j=1}^{k^2} (N_i)_{jj} C_j = r U_i, \quad i=1,2 \end{cases}$$

$c(U,N) = r^{-1}$ is the computed dilation constant.

7. Summary Table

Constant	Definition	Value for $d=2$
$C_2$	Universal commuting dilation constant	$\sqrt{2}$ (empirical)
Lower bound	$c(u_f, u_0)$	$\sqrt{2}$
Upper bound	$C_2 \leq 2 \sqrt{2/3} < 2$	Strictly $<2$

The collective evidence in (Gerhold et al., 14 Oct 2025) indicates that the universal commuting dilation constant for pairs of contractions is $\sqrt{2}$, with all operator tuples admitting commuting normal dilations of norm at most $\sqrt{2}$ times $\|T\|$. This development significantly narrows the gap in dilation theory, quantifies the noncommutative-vs-commutative divide, and sets the stage for further progress in understanding multivariable operator dilations.