  For Belavin's elliptic quantum R-matrix, we construct an L-operator as a set
of difference operators acting on functions on the type A weight space.
According to the fundamental relation $RLL=LLR$, the trace of the L-operator
gives a commuting difference operators. We show that for the above mentioned
L-operator this approach gives Macdonald type operators with elliptic theta
function coefficient, actually equivalent to Ruijsenaars' operators. The
relationship between the difference L-operator and Krichever's Lax matrix as
well as other approaches for the elliptic commuting system is given. We also
study the invariant subspace for the system which is spanned by symmetric theta
functions.
