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 $\mathcal{A}\subseteq \mathcal{B}(\mathcal{X})$ 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.

12 pages