Banach Journal of Mathematical Analysis

On existence of hyperinvariant subspaces for linear maps

Wieslaw Zelazko

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Let $X$ be an infinite dimensional complex vector space. We show that a non-constant endomorphism of $X$ has a proper hyperinvariant subspace if and only if its spectrum is non-void. As an application we show that each non-constant continuous endomorphism of the locally convex space $(s)$ of all complex sequences has a proper closed hyperinvariant subspace.

Article information

Banach J. Math. Anal. Volume 3, Number 1 (2009), 143-148.

First available in Project Euclid: 21 April 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A15: Invariant subspaces [See also 47A46]
Secondary: 15A04: Linear transformations, semilinear transformations

hyperinvariant subspace locally convex space endomorphism


Zelazko, Wieslaw. On existence of hyperinvariant subspaces for linear maps. Banach J. Math. Anal. 3 (2009), no. 1, 143--148. doi:10.15352/bjma/1240336431.

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  • S. Banach, Théorie des Opérations Linéaires, Warszawa 1932.
  • G. Köthe, Topological Vector Spaces I, Springer Verlag, 1969.
  • H.H. Schaefer, Eine Bemerkung zur Existenz invarianter Teilräume linearer Abbildungen, Math. Z. 82 (1963), 90.
  • H.H. Schaefer, Topological Vector Spaces, Springer Verlag, 1971.
  • W. Żelazko, Concerning closed invariant subspaces for endomorphisms of the space $(s)$, Periodica Math. Hungar. 44 (2002), 239–242.
  • W. Żelazko, Operators on locally convex spaces, in Operator Theory: Advances and Applications vol. 187, 237–247, 2008 Birkhauser Verlag Basel.