Asymptotic properties for compressions of two-isometries

Abstract

The bounded linear operators T on a Hilbert space H which have 2-isometric liftings S on K ⊃ H are investigated in this paper. We refer to  the structure of those operators in the case of general liftings, as well as for a more special type of such liftings S, namely for which H is  invariant to the orthogonal projection onto the kernel of S ∗S − I. In this special case we describe the canonical triangulation of T induced  by S, with entries expressed by operators close to contractions or isometries. Also, in this case we refer to some asymptotic properties of  T and of its adjoint T ∗ obtained by means of asymptotic limits associated to T ∗ and T, relative to a lifting S for T. The canonical  triangulations of T in this setting are described in detail, and as an application, the Wold type decomposition of a concave operator is  obtained.