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Quasinilpotency of commutators under a weakened Kleinecke--Shirokov condition
Abstract
Kramar and Kramar Fijavž proved that a commutator Y=AB−BA of two complex square matrices is nilpotent whenever Y commutes with an iterated commutator δYk(A), where δY(T)=YT−TY, and asked whether this result can be generalized to bounded operators on infinite-dimensional spaces. We show that for k≥ 2 such a commutator need not be quasinilpotent. On the other hand, for every k it is quasinilpotent whenever it is compact, or more generally whenever it is a Riesz operator or AY−YA is compact. Moreover, its Riesz projection at any non-zero isolated point of its spectrum always has infinite rank.
Related
BibTeX
@misc{stankovic2026quasinilpotency,
title = {{Quasinilpotency of commutators under a weakened Kleinecke--Shirokov condition}},
author = {Hranislav Stanković},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06423},
eprint = {2611.06423},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06423},
}
RIS
TY - UNPB
TI - Quasinilpotency of commutators under a weakened Kleinecke--Shirokov condition
AU - Hranislav Stanković
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06423
UR - https://arxiv.org/abs/2611.06423
AB - Kramar and Kramar Fijavž proved that a commutator Y=AB−BA of two complex square matrices is nilpotent whenever Y commutes with an iterated commutator δ_Y^k(A), where δ_Y(T)=YT−TY, and asked whether this result can be generalized to bounded operators on infinite-dimensional spaces. We show that for k≥ 2 such a commutator need not be quasinilpotent. On the other hand, for every k it is quasinilpotent whenever it is compact, or more generally whenever it is a Riesz operator or AY−YA is compact. Moreover, its Riesz projection at any non-zero isolated point of its spectrum always has infinite rank.
ER -