Hyperinvariant subspaces for operators having a compact part

It is well known that if T=A⊕B, where A is compact, then T has a nontrivial hyperinvariant subspace. In this paper, we try to solve the hyperinvariant subspace problem for operators which have a compact part. Our main result is that if A is compact, then either or has a nontrivial hyperinvariant subspace.