Commutators close to the identity

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Let D,X ∈ B(H) be bounded operators on an infinite dimensional Hilbert space H. If the commutator [D,X] = DX-XD lies within ε in operator norm of the identity operator 1B₍ₕ₎, then it was observed by Popa that one has the lower bound D X ≥ 1/2 1/ε on the product of the operator norms of D,X; this is a quantitative version of the Wintner-Wielandt theorem that 1B₍ₕ₎ cannot be expressed as the commutator of bounded operators. On the other hand, it follows easily from the work of Brown and Pearcy that one can construct examples in which D X = O(ε⁻²). In this note, we improve the Brown-Pearcy construction to obtain examples of D,X with [D,X] - 1B₍ₕ₎ ≤ ε and D X = O(⁵ 1/ε).