  In this paper we introduce preHamiltonian pairs of difference operators and
study their connections with Nijenhuis operators and the existence of weakly
non-local inverse recursion operators for differential-difference equations. We
begin with a rigorous setup of the problem in terms of the skew field $Q$ of
rational (pseudo--difference) operators over a difference field $F$ with a zero
characteristic subfield of constants $k\subset F$ and the principal ideal ring
$M_n(Q)$ of matrix rational (pseudo-difference) operators. In particular, we
give a criteria for a rational operator to be weakly non--local. A difference
operator $H$ is called preHamiltonian, if its image is a Lie $k$-subalgebra
with respect the the Lie bracket on $F$. Two preHamiltonian operators form a
preHamiltonian pair if any $k$-linear combination of them is preHamiltonian.
Then we show that a preHamiltonian pair naturally leads to a Nijenhuis
operator, and a Nijenhuis operator can be represented in terms of a
preHamiltonian pair. This provides a systematical method to check whether a
rational operator is Nijenhuis. As an application, we construct a
preHamiltonian pair and thus a Nijenhuis recursion operator for the
differential-difference equation recently discovered by Adler \& Postnikov. The
Nijenhuis operator obtained is not weakly non-local. We prove that it generates
an infinite hierarchy of local commuting symmetries. We also illustrate our
theory on the well known examples including the Toda, the Ablowitz-Ladik and
the Kaup-Newell differential-difference equations.
