Hilbert space operators with two-isometric dilations

TL;DR

This paper investigates Hilbert space operators that can be extended or lifted to 2-isometries, characterizing their adjoints and applying results to concave operators and those similar to contractions or isometries.

Contribution

It introduces new characterizations of operators with 2-isometric dilations and constructs minimal liftings, expanding understanding of operator dilations in Hilbert spaces.

Findings

Characterization of adjoints of operators with 2-isometric liftings

Construction of two types of liftings to 2-isometries

Application to concave operators and operators similar to contractions

Abstract

A bounded linear Hilbert space operator $S$ is said to be a $2$-isometry if the operator $S$ and its adjoint $S^*$ satisfy the relation $S^{*2}S^{2} - 2 S^{*}S + I = 0$. 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 $S^{*2}S^{2} - 2 S^{*}S + I \le 0$) 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.