Exponent for commutators close to the identity

Description of constant

Let $H$ be an infinite-dimensional complex Hilbert space and let $B (H)$ be the Banach algebra of bounded operators on $H$ , equipped with the operator norm. For $D, X \in B (H)$ let $[D, X] := D X - X D$ denote their commutator, and for $0 < ε < 1$ define $m_{H} (ε) := in f {∥ D ∥ ∥ X ∥ : D, X \in B (H), ∥[D, X] - 1_{B (H)} ∥ \leq ε} .$ The exponent for commutators close to the identity is $C_{85} := ε \to 0^{+} lim sup \frac{lo g m _{H} ( ε )}{lo g lo g ( 1/ ε )} .$ Equivalently, $C_{85}$ is the infimum of $c \geq 0$ for which one has $m_{H} (ε) ≪_{c} lo g^{c} (1/ ε)$ as $ε \to 0^{+}$ .

By a classical theorem of Wintner [Win47] and Wielandt [Wie49], the identity operator $1_{B (H)}$ cannot be written exactly as $[D, X]$ for bounded $D, X$ (indeed in any unital Banach algebra), but in infinite dimensions Brown and Pearcy [BP65] showed that it can be approximated arbitrarily well by bounded commutators, so $m_{H} (ε) < \infty$ for every $ε > 0$ . Popa [Pop82] gave a quantitative Wintner–Wielandt lower bound $m_{H} (ε) \geq \frac{1}{2} lo g (1/ ε)$ , so $C_{85} \geq 1$ . It is a conjecture (raised in [Pop82, Remark 2.9] and reiterated in [Tao19]) that Popa's exponent- $1$ bound is essentially sharp, i.e. $C_{85} = 1$ .

Known upper bounds

Bound	Reference	Comments
$5$	[Tao19]	Almost-upper-triangular matrices in $M_{n} (B (H))$ with $n ≍ lo g (1/ ε)$ , obtained by solving a nonlinear system in $B (H)^{n}$ via Cuntz isometries and a contraction-mapping argument. Tao's first-draft exponent was $16$ ; the anonymous referee reduced it to $5$ .
$5$	[KJ22]	Krishna–Johnson extend Tao's construction from $B (H)$ to unital $C^{\*}$ -algebras satisfying suitable structural hypotheses (in particular, algebras containing a unital copy of the Cuntz algebra $O_{2}$ ), with the same exponent. Does not improve $C_{85}$ but broadens the setting.
$4$	[Bil26]	Same nonlinear system as [Tao19], with a sharper point-source Green function estimate on the right inverse (norm $O (n)$ on scalar point sources, versus $O (n^{2})$ in the worst case), proved by interpreting iterated Cuntz block coefficients as a killed walk and comparing them with binomial incidence matrices. This upgrades the fixed-point scale from $δ ≍ n^{- 5}$ to $δ ≍ n^{- 4}$ and improves the exponent from $5$ to $4$ . Result and Lean 4 formalization produced autonomously by GPT-5.6 Sol High (13 min for the proof, $< 3$ h for the formalization).

Known lower bounds

Bound	Reference	Comments
$0$	Trivial	$m_{H} (ε) \geq 1$ for all $ε$ .
$1$	[Pop82]	Popa's quantitative Wintner–Wielandt bound $m_{H} (ε) \geq \frac{1}{2} lo g (1/ ε)$ ; see [Tao19, Theorem 0.1] for a short reproduction. Conjectured to be sharp.

Additional comments and links

The Brown–Pearcy construction [BP65], together with an elementary conjugation-by-diagonal step (Proposition 0.2 of [Tao19], attributed there to Popa), gives the earlier bound $m_{H} (ε) = O (ε^{- 2})$ . This is polynomial in $1/ ε$ and hence does not give a finite upper bound on $C_{85}$ ; polylogarithmic control in $1/ ε$ first appears in [Tao19].
The problem is intrinsically infinite-dimensional: if $dim H < \infty$ then any commutator $[D, X]$ has trace zero and therefore some eigenvalue outside the disk $z : ∣ z - 1 ∣ < 1$ , forcing $∥[D, X] - 1 ∥ \geq 1$ . Thus $m_{H} (ε) = + \infty$ for every $ε < 1$ in finite dimensions, and $C_{85}$ is unambiguously defined by the infinite-dimensional case.
A Lasserre / Navascués–Pironio–Acín semidefinite-programming approach to bounding $C_{85}$ from below was proposed in [Tao19, Remark 2.9] (an observation of Tobias Fritz); with computationally-feasible sets of noncommutative monomials it has so far not been observed to yield any nontrivial pairs $(C, ε)$ .
[Tao19, §2] also raises a related question in terms of the distance $dist (A, C + K (H))$ from $A \in B (H)$ to the scalar-plus-compact operators: whether the exponent $2/3$ in a result of [Pop82, Theorem 2.1] characterizing which $A$ are commutators of bounded operators can be replaced by a $lo g^{- C}$ factor.

References

[Win47] Wintner, Aurel. The unboundedness of quantum-mechanical matrices. Physical Review 71 (1947), 738–739.
[Wie49] Wielandt, Helmut. Über die Unbeschränktheit der Operatoren der Quantenmechanik. Mathematische Annalen 121 (1949), 21.
[BP65] Brown, Arlen; Pearcy, Carl. Structure of commutators of operators. Annals of Mathematics 82 (1965), 112–127.
[Pop82] Popa, Sorin. On commutators in properly infinite $W^{\*}$ -algebras. In: Invariant Subspaces and Other Topics (Timişoara/Herculane, 1981), Operator Theory: Advances and Applications 6, Birkhäuser, Basel, 1982, pp. 195–207.
[Tao19] Tao, Terence. Commutators close to the identity. Journal of Operator Theory 82 (2019), no. 2, 369–382. arXiv:1805.11131.
[KJ22] Krishna, K. Mahesh; Johnson, P. Sam. Commutators close to the identity in unital $C^{\*}$ -algebras. Proceedings — Mathematical Sciences 132 (2022), Paper No. 11. DOI: 10.1007/s12044-022-00663-w. arXiv:2104.02035.
[Bil26] Bilich, Boris. An $O (lo g^{4} (1/ ε))$ refinement of Tao's construction of commutators close to the identity. Note (2026); result and Lean 4 formalization produced autonomously by GPT-5.6 Sol High. Available at bilichboris.github.io/blog/2026/gpt-5-6-sol-improved-tao-bound (accessed 17 Jul 2026).

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.