Hyperinvariant subspaces for operators having a compact part

It is well known that if $T=A\oplus B$, 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 $\left(\begin{array}{cc}A\hfill & \hfill ?\hfill \\ 0\hfill & \hfill B\hfill \end{array}\right)$ or $\left(\begin{array}{cc}A\hfill & \hfill 0\hfill \\ ?\hfill & \hfill B\hfill \end{array}\right)$ has a nontrivial hyperinvariant subspace.