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∈B(H) let [D,X]:=DX−XD denote their commutator, and for 0<ε<1 define
mH(ε):=inf{∥D∥ ∥X∥:D,X∈B(H), ∥[D,X]−1B(H)∥≤ε}.
The exponent for commutators close to the identity is
C85:=ε→0+limsuploglog(1/ε)logmH(ε).
Equivalently, C85 is the infimum of c≥0 for which one has mH(ε)≪clogc(1/ε) as ε→0+.
By a classical theorem of Wintner [Win47] and Wielandt [Wie49], the identity operator 1B(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 mH(ε)<∞ for every ε>0. Popa [Pop82] gave a quantitative Wintner–Wielandt lower bound mH(ε)≥21log(1/ε), so C85≥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. C85=1.