Hyperinvariant Subspaces of Weak Contractions. II.

Abstract

This note is a continuation of [6]. We shall use the terminology and notation of [6]. Recall that an operator T on a (complex separable) Hilbert space H is said to have the property (L) if Hyplat(T) is the smallest complete lattice containing all subspaces of the forms Ker S and $\overline {{\mathbf{Ran}}} {\text{ S}} = \overline {{\text{SH}}}$ , for S ε {T}″. It was proved in [5], [6], that all c.n.u. weak contractions have the property (L). Recently it was shown [7] that an isometry V has the property (L) if and only if either V is unitary or the absolutely continuous unitary part of V is zero. In particular every c.n.u. isometry has (L), but not all isometries have the property (L). The purpose of this note is to show that all (not necessarily c.n.u.) weak contractions have the property (L). To do this we shall need a generalization of two lemmas due to L. Kérchy [3].