Banach space operators acting on some fixed space X are considered. If two such operators A and B verify the condition A2 = B2 and if A has non-trivial hyperinvariant subspaces, then B has nontrivial invariant subspaces. If A and B commute and satisfy a special type of functional equation, and if A is not a scalar multiple of the identity, the author proves that if A has nontrivial invariant subspaces, then so does B.
Matache, Valentin, "Operator Equations and Invariant Subspaces" (1994). Mathematics Faculty Publications. 28.