On commutators of square-zero Hilbert space operators

TL;DR

This work analyzes the set $\mathfrak c({nil}_2)=\{ MN-NM : M^2=N^2=0\}$ of commutators of square-zero operators on a complex Hilbert space. It delivers a complete finite-dimensional description of $\mathfrak c({nil}_2)$ and its norm-closure, and investigates the infinite-dimensional case by connecting the closure to biquasitriangular operators via Bal-type invariants and revealing an index-based obstruction to clos membership. Structurally, the authors reduce questions to $T\sim A\oplus B$ with disjoint $\sigma(A^2)$ and $\sigma(B^2)$, showing invertible blocks must be similar to $A_0\oplus(-A_0)$ and that normal $T$ in $\mathfrak c({nil}_2)$ are unitarily equivalent to $-T$. In finite dimensions, the nilpotent part must be a square of a nilpotent operator, characterized through Borwein–Richmond, yielding concrete Jordan-form criteria. They show $clos(\mathfrak c({nil}_2))=Bal(\mathcal H)$ in finite dimensions and provide spectrum-based obstructions and biquasitriangular intersections in infinite dimensions, along with a concrete obstruction for Fredholm operators with odd Fredholm index. Overall, the paper advances understanding of commutator structures for nilpotent-like operators and clarifies how similarity-invariant closures behave in both finite and infinite settings.

Abstract

Let $\mathcal{H}$ be a complex, separable Hilbert space, and set $\mathfrak{c}($NIL$_2)=\{ MN - NM : N, M \in \mathcal{B}(\mathcal{H}), M^2 = 0 = N^2 \}$. When $\dim\, \mathcal{H}$ is finite, we characterise the set $\mathfrak{c}($NIL$_2)$ and its norm-closure CLOS$(\mathfrak{c}($NIL$_2))$. In the infinite-dimensional setting, we characterise the intersection of CLOS$(\mathfrak{c}($NIL$_2))$ with the set of biquasitriangular operators, and we exhibit an index obstruction to belonging to CLOS$(\mathfrak{c}($NIL$_2))$.