Hilbert space operators with two-isometric dilations

A continuous linear Hilbert space operator S is said to be a 2-isometry if the and its adjoint S∗ satisfy relation S∗2S2−2S∗S+I=0. We study operators having liftings or dilations 2-isometries. The of an which admits such restriction backward shift on vector-valued analytic functions. These results are applied concave similar contractions. Two types 2-isometries, as well extensions induced by them, constructed isomorphic minimal discussed.