Three-isometric liftings with invariant isometric part

Abstract

In this article we refer to operators T on a Hilbert space $\mathcal {H}$ which can be lifted to some 3-isometries S on Hilbert spaces $\mathcal {K}$ containing $\mathcal {H}$ (as a closed subspace). Namely, we consider only liftings S with the invariant part $\mathcal {K}\ominus \mathcal {H}$ included in the kernel of the covariance operator $S^*S-I$. Such operators T can be identified as generalized (A, 2)-isometries, for self-adjoint operators A on $\mathcal {H}$ with $A\ge T^*T-I$. The case when A is positive corresponds exactly to an expansive lifting S for T, and in this context are mentioned the operators similar to 2-isometries. We also describe such operators T by different triangulations induced by their liftings S. Here the entries are contractions, or operators intertwined with isometries by some positive operators depending on T. In the case when T is an (A, 2)-isometry and A-bounded, such a triangulation is obtained under the decomposition $\mathcal {H}=\mathcal {N}(A)\oplus \overline{\mathcal {R}(A)}$. Here the compression of T on $\overline{\mathcal {R}(A)}$ is intertwined with a 2-isometry.