Complex Analysis and Operator Theory (2020) 14:5 https://doi.org/10.1007/s11785-019-00960-9 Complex Analysis and Operator Theory On Operators with Two-Isometric Liftings Laurian Suciu 1 Received: 16 June 2019 / Accepted: 13 December 2019 © Springer Nature Switzerland AG 2020 Abstract We investigate the operators T on a Hilbert space H which have 2-isometric liftings S on K ⊃ H such that H is an invariant subspace for S ∗ S. We describe such operators by a special lifting S for which the covariance operator  S = S ∗ S − I on K ⊖ H is a scalar multiple of an orthogonal projection. Also, we characterize the minimal liftings S for T in the terms of operators from the block matrix of S which contains T . An upper triangulation of some operators T with 2-isometric liftings is obtained, where the restriction of T to the corresponding invariant subspace is a contraction. Other sufficient conditions for such triangular operators to have 2-isometric liftings are given. Keywords 2-Isometric lifting · Covariance · A-contraction Mathematics Subject Classification 47A05 · 47A15 · 47A20 · 47A63 1 Introduction and Preliminaries The Sz.-Nagy-Foias dilation theory of Hilbert space contractions is well-known, where the isometric liftings and the unitary dilations have an essential role (see [10,20]). In the last years, a class of bounded linear operators closely related to isometries, namely the 2-isometries, were intensively studied by many authors (see, for instance, [1– 5,8,11,13–16]). A 2-isometry is an operator T on a (complex) Hilbert space H which satisfies the second order difference condition ‖T 2 h ‖ 2 − 2‖Th ‖ 2 +‖h ‖ 2 = 0, h ∈ H. (1.1) Communicated by Vladimir Bolotnikov. B Laurian Suciu laurians2002@yahoo.com 1 Department of Mathematics and Informatics, “Lucian Blaga” University of Sibiu, Sibiu, Romania 0123456789().: V,-vol