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On almost commuting matrices with respect to the normalized Hilbert--Schmidt norm
Abstract
In this paper, we consider the Rosenthal -- Halmos problem of almost commuting matrices with respect to the normalized Hilbert -- Schmidt norm |·|2,d=d−1/2|·|2. We show that if X and Y are self-adjoint matrices with |X|≤ 1, |Y|≤ 1, then there exist commuting self-adjoint matrices X',Y' such that |X−X'|2,d+|Y−Y'|2,d≤ 5|[X,Y]|2,d1/3, and [X,X']=0. Within these constraints, the exponent 1/3 cannot be improved.
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BibTeX
@misc{bansil2026almost,
title = {{On almost commuting matrices with respect to the normalized Hilbert--Schmidt norm}},
author = {Mohit Bansil and Ilya Kachkovskiy},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.31000},
eprint = {2608.31000},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.31000},
}
RIS
TY - UNPB
TI - On almost commuting matrices with respect to the normalized Hilbert--Schmidt norm
AU - Mohit Bansil
AU - Ilya Kachkovskiy
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.31000
UR - https://arxiv.org/abs/2608.31000
AB - In this paper, we consider the Rosenthal -- Halmos problem of almost commuting matrices with respect to the normalized Hilbert -- Schmidt norm |·|_2,d=d^−1/2|·|_2. We show that if X and Y are self-adjoint matrices with |X|≤ 1, |Y|≤ 1, then there exist commuting self-adjoint matrices X',Y' such that |X−X'|_2,d+|Y−Y'|_2,d≤ 5|[X,Y]|_2,d^1/3, and [X,X']=0. Within these constraints, the exponent 1/3 cannot be improved.
ER -