On the Commutant of Invertible Positive Operators
Abstract
An important result due to Radjabalipour asserts that if A∈BB(H) is an invertible positive operator and T∈BB(H) is arbitrary, then the operator sequence {A−nTAn}n∈ZZ is norm-bounded if and only if TA=AT. Using the concept of spectral gaps, we show that if A possesses a spectral gap and the sequence {A−nTAn}n∈ZZ is norm-bounded, then T must be block diagonal with respect to a certain orthogonal decomposition of H. We also provide an alternative proof of Radjabalipour's theorem. We also extend the theorem to some rich classes of operators, including invertible normal operators and invertible weighted sums of projections. Moreover, we provide simple proofs for the fact that a positive definite matrix A∈MMN is a positive multiple of the identity if and only if the sequence {A−nTAn}n∈ZZ is norm-bounded for every~T∈MMN.
Related
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- The Keller problem for the lowest positive eigenvalue of Dirac operators with Lorentz scalar potentials
BibTeX
@misc{kittaneh2026commutant,
title = {{On the Commutant of Invertible Positive Operators}},
author = {Fuad Kittaneh and Carlos S. Kubrusly and Mohammad Sal Moslehian},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21023},
eprint = {2609.21023},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21023},
}
RIS
TY - UNPB
TI - On the Commutant of Invertible Positive Operators
AU - Fuad Kittaneh
AU - Carlos S. Kubrusly
AU - Mohammad Sal Moslehian
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.21023
UR - https://arxiv.org/abs/2609.21023
AB - An important result due to Radjabalipour asserts that if A∈BB(H) is an invertible positive operator and T∈BB(H) is arbitrary, then the operator sequence {A^−nTA^n}_n∈ZZ is norm-bounded if and only if TA=AT. Using the concept of spectral gaps, we show that if A possesses a spectral gap and the sequence {A^−nTA^n}_n∈ZZ is norm-bounded, then T must be block diagonal with respect to a certain orthogonal decomposition of H. We also provide an alternative proof of Radjabalipour's theorem. We also extend the theorem to some rich classes of operators, including invertible normal operators and invertible weighted sums of projections. Moreover, we provide simple proofs for the fact that a positive definite matrix A∈MM_N is a positive multiple of the identity if and only if the sequence {A^−nTA^n}_n∈ZZ is norm-bounded for every~T∈MM_N.
ER -