Hyperinvariant subspaces for some subnormal operators

Abstract

In this article we employ a technique originated by Enflo in 1998 and later modified by the authors to study the hyperinvariant subspace problem for subnormal operators. We show that every “normalized” subnormal operator S S such that either { ( S ∗ n S n ) 1 / n } \{(S^{\ast n}S^{n})^{1/n}\} does not converge in the SOT to the identity operator or { ( S n S ∗ n ) 1 / n } \{(S^{n}S^{\ast n})^{1/n}\} does not converge in the SOT to zero has a nontrivial hyperinvariant subspace.