PreprintCC BY 4.0arXiv
Geometric operators and the third problem of Halmos
Abstract
In 1970, P. R. Halmos raised ten open problems on operator theory. The third problem of Halmos is the following: if an intransitive operator has an inverse, is its inverse also intransitive? This question is reduced to the case of conjugate bi-geometric operators in arXiv: 2610.11202. In the framework of conjugate bi-geometric operators, using tools from complex geometry, operator theory, and function theory, this paper gives an affirmative answer to the third problem of Halmos in the following cases: (a) bundle-preserving operator;(b) subexponential growth of operator norms. These two cases cover a large class of operators.
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BibTeX
@misc{ji2026geometric,
title = {{Geometric operators and the third problem of Halmos}},
author = {Kui Ji and Chunlan Jiang and Zhenyue Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12857},
eprint = {2610.12857},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12857},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometric operators and the third problem of Halmos
AU - Kui Ji
AU - Chunlan Jiang
AU - Zhenyue Wang
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.12857
UR - https://arxiv.org/abs/2610.12857
AB - In 1970, P. R. Halmos raised ten open problems on operator theory. The third problem of Halmos is the following: if an intransitive operator has an inverse, is its inverse also intransitive? This question is reduced to the case of conjugate bi-geometric operators in arXiv: 2610.11202. In the framework of conjugate bi-geometric operators, using tools from complex geometry, operator theory, and function theory, this paper gives an affirmative answer to the third problem of Halmos in the following cases: (a) bundle-preserving operator;(b) subexponential growth of operator norms. These two cases cover a large class of operators.
ER -