Commutators close to the identity

TL;DR

This paper improves the construction of operator pairs whose commutator approximates the identity, showing that their norms can be bounded by a logarithmic function of the approximation error, refining previous polynomial bounds.

Contribution

It provides a new construction demonstrating that the product of norms of operators approximating a commutator to the identity can be bounded by a logarithmic function of the inverse error.

Findings

Constructed operators with commutator close to identity and bounded norms

Bounded the product of operator norms by a logarithmic function of the inverse error

Improved previous polynomial bounds to logarithmic bounds

Abstract

Let $D,X \in B(H)$ be bounded operators on an infinite dimensional Hilbert space $H$. If the commutator $[D,X] = DX-XD$ lies within $\varepsilon$ in operator norm of the identity operator $1_{B(H)}$, then it was observed by Popa that one has the lower bound $\| D \| \|X\| \geq \frac{1}{2} \log \frac{1}{\varepsilon}$ on the product of the operator norms of $D,X$; this is a quantitative version of the Wintner-Wielandt theorem that $1_{B(H)}$ 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 $\|D\| \|X\| = O(\varepsilon^{-2})$. In this note, we improve the Brown-Pearcy construction to obtain examples of $D,X$ with $\| [D,X] - 1_{B(H)} \| \leq \varepsilon$ and $\| D\| \|X\| = O( \log^{5} \frac{1}{\varepsilon} )$.