High order isometric liftings and dilations

We show that a Hilbert space bounded linear operator has an $m$-isometric lifting for some integer $m\ge 1$ if and only if the norms of its powers grow polynomially. In analogy with unitary dilations of contractions, we prove that such operators also have an invertible $m$-isometric dilation. We also study $2$-isometric liftings of convex operators and $3$-isometric liftings of Foguel-Hankel operators.