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 $[D,X] := DX - XD$ denote their commutator, and for $0 < \varepsilon < 1$ define \(m_H(\varepsilon) := \inf \\{ \lVert D\rVert\ \lVert X\rVert : D, X \in B(H),\ \lVert [D,X] - 1\_{B(H)}\rVert \le \varepsilon \\}.\) The exponent for commutators close to the identity is \(C_{85} := \limsup_{\varepsilon \to 0^+} \frac{\log m_H(\varepsilon)}{\log \log(1/\varepsilon)}.\) Equivalently, $C_{85}$ is the infimum of $c \ge 0$ for which one has $m_H(\varepsilon) \ll_c \log^c(1/\varepsilon)$ as $\varepsilon \to 0^+$.
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 $m_H(\varepsilon) < \infty$ for every $\varepsilon > 0$. Popa [Pop82] gave a quantitative Wintner–Wielandt lower bound $m_H(\varepsilon) \ge \tfrac{1}{2}\log(1/\varepsilon)$, so $C_{85} \ge 1$. 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 $n \asymp \log(1/\varepsilon)$, 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 $\mathcal{O}_2$), 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 $\delta \asymp n^{-5}$ to $\delta \asymp n^{-4}$ 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 | $m_H(\varepsilon) \ge 1$ for all $\varepsilon$. |
| $1$ | [Pop82] | Popa’s quantitative Wintner–Wielandt bound $m_H(\varepsilon) \ge \tfrac{1}{2}\log(1/\varepsilon)$; 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 $m_H(\varepsilon) = O(\varepsilon^{-2})$. This is polynomial in $1/\varepsilon$ and hence does not give a finite upper bound on $C_{85}$; polylogarithmic control in $1/\varepsilon$ 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 $\{z : \lvert z-1\rvert < 1\}$, forcing $\lVert [D,X] - 1\rVert \ge 1$. Thus $m_H(\varepsilon) = +\infty$ for every $\varepsilon < 1$ 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 $(C,\varepsilon)$.
- [Tao19, §2] also raises a related question in terms of the distance $\mathrm{dist}(A, \mathbb{C} + K(H))$ 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 $\log^{-C}$ 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 $O(\log^4(1/\varepsilon))$ 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).
Contribution notes
Prepared with assistance from Claude Opus 4.7, which read the [Tao19] and [Bil26] preprints, retrieved [KJ22] via web search, and drafted the page. All references should be independently verified before citation.