Lifting commuting 3-isometric tuples

An operator T is called a 3-isometry if there exists operators B₁(T*,T) and B₂(T*,T) such that Q(n) = T*ⁿTⁿ = 1+nB₁(T*,T)+n²B₂(T*,T) for all natural numbers n. An operator J is a Jordan operator of order 2 if J = U +N where U is unitary, N is nilpotent order 2, and U and N commute. An easy computation shows that J is a 3-isometry and that the restriction of J to an invariant subspace is also a 3-isometry. Those 3-isometries which are the restriction of a Jordan operator to an invariant subspace can be identified, using the theory of completely positive maps, in terms of a positivity condition on the operator pencil Q(s). In this article, we establish the analogous result in the multi-variable setting and show, by modifying an example of Choi, that an additional hypothesis is necessary. Lastly we discuss the joint spectrum of sub-Jordan tuples and derive results for 3-symmetric operators as a corollary.