Triangulations of operators with two-isometric liftings
The paper deals with bounded linear operators on a complex Hilbert space which have 2-isometric liftings. Recall that a bounded linear operator \(T\) on \(H\) is called a 2-isometry if \(I-2T^*T+T^{*2}T^2=0\). There is a special class of \(2\)-isometric liftings, namely, those liftings which satisfy the range inclusion of the covariance operators. Here, the covariance operator for \(T\) is given by \(\Delta_T=T^*T-I\). If \(T\) is a \(2\)-isometry, then \(\Delta_T\) is a positive operator, \(\ker \Delta_T\) is an invariant subspace for \(T\) and \(T|_{\ker \Delta_T}\) is an isometry. One of the main results of the paper under review classifies 2-isometric liftings \(S\) on \(K\) of a \(2\)-isometry \(T\) on \(H\) which satisfy the condition \[ \operatorname{Range}(\Delta_{S|_{K \ominus H}}) \subseteq \overline{\operatorname{Range}(\Delta_{S})} \] (see Theorem 2.3 for a precise statement). The authors also discuss various types of \(2\)-isometric liftings and state some open problems pertaining to classifications of these liftings.
scientific article; zbMATH DE number 7345255
scientific article