  In this paper we prove that if T:C[0,1] \rightarrow C[0,1] is a positive
linear operator with T(e_0)=1 and T(e_1)-e_1 does not change the sign, then the
iterates T^{m} converges to some positive linear operator T^{\infty} :C[0,1]
\rightarrow C[0,1] and we derive quantitative estimates in terms of modulii of
smoothness. This result enlarges the class of operators for which the limit of
the iterates can be computed and the quantitative estimates of iterates can be
given.
