On operators with two-isometric liftings (Q2299287)

Let \(\mathcal H\) and \(\mathcal K\) be complex Hilbert spaces. A 2-isometry is a Hilbert-space operator \(S\) such that \({\|S^2x\|^2-2\|Sx\|^2+\|x\|^2=0}\) for every \({x\in\mathcal K}\). The paper deals with Hilbert-space operators \(T\) on \(\mathcal H\) that can be lifted to a 2-isometry \(S\) on a Hilbert space \(\mathcal K\) including \(\mathcal H\) (i.e., operators \(T\) such that \({PS=TP}\) where \(P\) is the orthogonal projection on \(\mathcal K\) with range equal to \(\mathcal H\)). It investigates the case where \({S^*S(\mathcal H)\subseteq\mathcal H}\) (i.e., where \(\mathcal H\) is an invariant subspace for \({|S|^{1/2}=S^*S}\)). The main results involve the characterization of a 2-isometric lifting \(S\) in terms of its defect operator \({I-S^*S}\) (such that \({(I-S^*S)|_{\mathcal K\ominus\mathcal H}}\) is a scalar multiple of an orthogonal projection -- cf.\ Theorem 2.1.). The minimal lifting problem is also considered, and relevant corollaries are harvested. The paper is carefully written and well-organized bringing interesting results, including upper triangulation of operators with a 2-isometric lifting.