The operators T on a Hilbert space \({\mathcal {H}}\) which have 3-isometric liftings S on Hilbert spaces \({\mathcal {K}}\) containing \({\mathcal {H}}\) are investigated. Here, we deal with the liftings S which are 2-isometries on their invariant subspace \({\mathcal {K}}\ominus {\mathcal {H}}\) and also have this subspace invariant for \(S^*S\) . Several characterizations for such operators T are obtained, including the case when the lifting S is expansive. As an application, we refer to the class of (A, 2)-contractions for a positive operator A on \({\mathcal {H}}\) , and particularly to the expansive 3-concave operators. Also, we obtain an upper triangulation for T induced by S, where the operators on the main diagonal are close to contractions. Finally, we study such liftings S which are minimal, in the sense that \({\mathcal {K}}\) is generated by \(\{S^n{\mathcal {H}},n\ge 0\}\) .
