Spectral Radius Algebras of Idempotents

We show that the spectral radius algebras of certain quadratic operators possess nontrivial invariant subspaces. Additionally, such algebras properly contain the operator’s commutant, so that the invariant subspaces are in some sense beyond hyperinvariant. The spectral radius algebras of idempotents are completely described and, as a consequence, it is shown that every intransitive collection of operators must be contained in a norm-closed proper spectral radius algebra.