Operators with a non-trivial closed invariant affine subspace

Bračič, Janko (author)

We are concerned with the question of the existence of an invariant proper affine subspace for an operator $A$ on a complex Banach space. It turns out that the presence of the number $1$ in the spectrum of $A$ or in the spectrum of its adjoint operator $A^*$ is crucial. For instance, an algebraic operator has an invariant proper affine subspace if and only if $1$ is its eigenvalue. For an arbitrary operator $A$, we show that it has an invariant proper hyperplane if and only if $1$ is an eigenvalue of $A^*$. If $A$ is a power bounded operator, then every invariant proper affine subspace is contained in an invariant proper hyperplane, moreover, $A$ has a non-trivial invariant cone.

• Language: English
• Keywords: invariant subspaces, invariant affine subspaces, power bounded operators
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: NTF - Faculty of Natural Sciences and Engineering
• Publication status: Published
• Publication version: Version of Record
• Year: 2024
• Number of pages: Str. 1305–1315
• Numbering: Vol. 98, iss. 5
• PID: 20.500.12556/RUL-166410
• UDC: 517.9
• ISSN on article: 0001-9054
• DOI: 10.1007/s00010-024-01090-0
• COBISS.SI-ID: 221338371
• Publication date in RUL: 10.01.2025

Record is a part of a journal

• Title: Aequationes mathematicae
• Shortened title: Aequ. math.
• Publisher: Springer
• ISSN: 0001-9054
• COBISS.SI-ID: 1327364

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: invariantni podprostori, invariantni afini podprostori, potenčno omejeni operatorji

Projects

• Funder: ARIS - Slovenian Research and Innovation Agency
• Project number: P2-0268
• Name: Geotehnologija