On existence of hyperinvariant subspaces for linear maps
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 \(\sigma(T)\neq\emptyset\). As an application, it is shown that \(T'\) is transitive iff either \(T=\alpha I\) for some \(\alpha\in\mathbb C\), 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.
scientific article; zbMATH DE number 5379957