A note about maximal almost-invariant subspaces and maximal hyperinvariant subspaces

Abstract

In this paper, we show that for T ∈ B(H), if M is almost-invariant for T , then every maximal almost-invariant subspace of M is of codimension 1 in M, where H is a separable, infinite-dimensional Hilbert space. We also describe the maximal hyperinvariant subspaces for normal operators with all the dimensions of eigenspaces at most 1 acting on H. Our result is that for each hyperinvariant subspace, all its maximal hyperinvariant subspaces are also of codimension 1 in it.