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arXiv · 2402.15676

On commutators of square-zero Hilbert space operators

Laurent W. Marcoux·Heydar Radjavi·Yuanhang Zhang
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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$_2))$.

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Laurent W. Marcoux, Heydar Radjavi, Yuanhang Zhang. 2024-02-24. On commutators of square-zero Hilbert space operators. https://doi.org/10.4153/s0008414x24000750

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