Optimal lengths of commutators of idempotents and similarity of nilpotent operators to their negatives
Abstract
Let HH be a complex separable infinite-dimensional Hilbert space. We prove that every operator in BH is a sum of three commutators of idempotents and a product of two such commutators. Both numbers are best possible. The same is true for differences of idempotents, for operators similar to their negatives, and for operators anticommuting with an involution. Each of these four classes therefore has additive length three and multiplicative length two. We also construct nilpotent operators of order three that are similar to their negatives but cannot be conjugated to their negatives by any bounded involution. These results answer Question~7.22(a)--(c), the question in Remark~7.17, and the question about involutions in Section~2.9 of Marcoux, Radjavi and Zhang, \emph{J. Funct. Anal.} \textbf{284} (2023), 109854.
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BibTeX
@misc{aouichaoui2026optimal,
title = {{Optimal lengths of commutators of idempotents and similarity of nilpotent operators to their negatives}},
author = {Mohamed Amine Aouichaoui},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01316},
eprint = {2611.01316},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01316},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal lengths of commutators of idempotents and similarity of nilpotent operators to their negatives
AU - Mohamed Amine Aouichaoui
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01316
UR - https://arxiv.org/abs/2611.01316
AB - Let HH be a complex separable infinite-dimensional Hilbert space. We prove that every operator in BH is a sum of three commutators of idempotents and a product of two such commutators. Both numbers are best possible. The same is true for differences of idempotents, for operators similar to their negatives, and for operators anticommuting with an involution. Each of these four classes therefore has additive length three and multiplicative length two. We also construct nilpotent operators of order three that are similar to their negatives but cannot be conjugated to their negatives by any bounded involution. These results answer Question~7.22(a)--(c), the question in Remark~7.17, and the question about involutions in Section~2.9 of Marcoux, Radjavi and Zhang, \emph{J. Funct. Anal.} \textbf{284} (2023), 109854.
ER -