  An operator $T$ is called a 3-isometry if there exists operators $B_1(T^*,T)$
and $B_2(T^*,T)$ such that \[Q(n)=T^{*n}T^n=1+nB_1(T^*,T)+n^2 B_2(T^*,T)\] 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.
