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Zbl 1171.47004
Żelazko, Wieslaw
On existence of hyperinvariant subspaces for linear maps.
(English)
[J] Banach J. Math. Anal. 3, No. 1, 143-148, electronic only (2009). ISSN 1735-8787/e

Let $X$ be an infinite-dimensional complex vector space and let $T\in L(X)$ be non-constant. In this paper, it is proved that $T$ has a proper hyperinvariant subspace iff $\sigma(T)\ne\emptyset$. As an application, it is shown that $T'$ is transitive iff either $T=\alpha I$ for some $\alpha\in\Bbb C$, or $\sigma(T)=\emptyset$. Also, it is proved that if $X$ is the complex space of all sequences, then $T$ has a proper closed hyperinvariant subspace.
[Bahmann Yousefi (Shiraz)]
MSC 2000:
*47A15 Invariant subspaces of linear operators
15A04 Linear transformations (linear algebra)

Keywords: hyperinvariant subspace; locally convex space; endomorphism

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