Characteristic subspaces and hyperinvariant frames

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|K|=2, since all characteristic subspaces are hyperinvariant if |K|>2|K|>2. The hyperinvariant hull WhWh of a subspace W of V is defined to be the smallest hyperinvariant subspace of V that contains W  , the hyperinvariant kernel WHWH of W is the largest hyperinvariant subspace of V that is contained in W  , and the pair (WH,Wh)(WH,Wh) 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 [WH,Wh][WH,Wh] are characteristic. We use this result for the construction of characteristic non-hyperinvariant subspaces.