Hyperinvariant subspaces for sets of polynomially compact operators

TL;DR

This paper proves the existence of non-trivial hyperinvariant subspaces for certain sets of polynomially compact operators, advancing understanding of their structure in operator theory.

Contribution

It establishes the existence of hyperinvariant subspaces for polynomially compact operators within specific algebraic and closure conditions, a novel result in the field.

Findings

Polynomially compact quasinilpotent operators have non-trivial hyperinvariant subspaces.

Algebras with polynomially compact operators contain non-trivial hyperinvariant subspaces.

Operator bands with a non-zero compact operator in their closure have hyperinvariant subspaces.

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.