Hilbert space operators with two-isometric dilations

Abstract

A bounded linear Hilbert space operator $S$ is said to be a $2$-isometry if the operator $S$ and its adjoint $S^{\ast }$ satisfy the relation $S^{\ast 2}{S}^2-2S^{\ast }S+I=0$. In this paper, we study Hilbert space operators having liftings or dilations to $2$-isometries. The adjoint of an operator which admits such liftings is characterized as the restriction of a backward shift on a Hilbert space of vector-valued analytic functions. These results are applied to concave operators (i.e., operators $S$ such that $S^{\ast 2}{S}^2-2S^{\ast }S+I\le 0$) and to operators similar to contractions or isometries. Two types of liftings to $2$-isometries, as well as the extensions induced by them, are constructed and isomorphic minimal liftings are discussed.

30 pages ; to appear in J. Operator Th