Characteristic subspaces and hyperinvariant frames

Abstract

Let f be an endomorphism of a finite dimensional vector space V over a field K. An f -invariant subspace is called hyperinvariant (respectively characteristic) if it is invariant under all endomorphisms (respectively automorphisms) that commute with f . We assume |K| = 2, since all characteristic subspaces are hyperinvariant if |K| > 2. The hyperinvariant hull W h of a subspace W of V is defined to be the smallest hyperinvariant subspace of V that contains W , the hyperinvariant kernel W H of W is the largest hyperinvariant subspace of V that is contained in W , and the pair (W H , W h ) is the hyperinvariant frame of W . In this paper we study hyperinvariant frames of characteristic non-hyperinvariant subspaces W . We show that all invariant subspaces in the interval [W H , W h ] are characteristic. We use this result for the construction of characteristic non-hyperinvariant subspaces.Mathematical Subject Classifications (2010): 15A04, 47A15, 15A18, 20K01.