The invariant subspace problem is a partially unresolved problem in functional analysis asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself. Many variants have been solved by restricting the class of operators or the class of Banach spaces considered, but the problem remains open for separable Hilbert spaces. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementWhether every bounded linear operator on a complex Banach space of dimension greater than one has a nontrivial closed invariant subspace; resolved for many restricted classes of operators and spaces, but open in general for separable Hilbert spaces. 1 Progress Toward ResolutionNo proof has been generally accepted by the mathematical community. In May 2023 a preprint by Per Enflo claimed to resolve the problem for Hilbert spaces, and in July 2023 a second, independent preprint by Neville claimed a solution for separable Hilbert spaces. 1 Classification
Resolution Status Resolution Status Prize Status
Prize Status (category) Connections
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Source Invariant Subspace Problem (Wikipedia)
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1. Invariant Subspace Problem (Wikipedia)
Wikimedia FoundationLead section
the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself
Lead section, first sentence
the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself
History section, final paragraph
In May 2023, a preprint of Enflo appeared on arXiv, which, if correct, solves the problem for Hilbert spaces and completes the picture.
View the Source Invariant Subspace Problem (Wikipedia)
WikipediaIn Branch: Functional Analysis, Lead sentenceQuote, In Branch: Functional Analysis, Lead sentence
In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking
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