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An operator T on Hilbert space is a 3-isometry if there exists operators Band D such that (T*)^n T^n = I+nB +n^2 D. An operator J is a Jordan operator ifit the sum of a unitary U and nilpotent N of order two which commute. If T is a3-isometry and c>0, then I-c^{-2} D + sB + s^2D is positive semidefinite forall real s if and only if T is the restriction to an invariant subspace of aJordan operator J=U+N with the norm of N at most c. As a corollary, ananalogous result for 3-symmetric operators, due to Helton and Agler, isrecovered.

A lifting theorem for 3-isometries