Hyperinvariant subspaces for sets of polynomially compact operators

Abstract

We prove the existence of a non-trivial hyperinvariant subspace for several sets of polynomially compact operators. The main results of the paper are: (i) a non-trivial norm closed algebra \(\mathcal {A}\subseteq \mathcal {B}(\mathscr {X})\) which consists of polynomially compact quasinilpotent operators has a non-trivial hyperinvariant subspace; (ii) if there exists a non-zero compact operator in the norm closure of the algebra generated by an operator band \(\mathcal {S}\), then \(\mathcal {S}\) has a non-trivial hyperinvariant subspace.