  Kantor pairs, (quadratic) Jordan pairs, and similar structures have been
instrumental in the study of $\mathbb{Z}$-graded Lie algebras and algebraic
groups. We introduce the notion of an operator Kantor pair, a generalization of
Kantor pairs to arbitrary (commutative, unital) rings, similar in spirit as to
how quadratic Jordan pairs and algebras generalize linear Jordan pairs and
algebras. Such an operator Kantor pair is formed by a pair of $\Phi$-groups
$(G^+,G^-)$ of a specific kind, equipped with certain homogeneous operators.
For each such a pair $(G^+,G^-)$, we construct a $5$-graded Lie algebra $L$
together with actions of $G^\pm$ on $L$ as automorphisms. Moreover, we can
associate a group $G(G^+,G^-) \subset \operatorname{Aut}(L)$ to this pair
generalizing the projective elementary group of Jordan pairs. If the
non-$0$-graded part of $L$ is projective, we can uniquely recover $G^+,G^-$
from $G(G^+,G^-)$ and the grading on $L$ alone. We establish, over rings $\Phi$
with $1/30 \in \Phi$, a one to one correspondence between Kantor pairs and
operator Kantor pairs. Finally, we construct operator Kantor pairs for the
different families of central simple structurable algebras.
