Constrictive Markov operators induced by Markov processes

Consider a constrictive Markov operator (T:L^1(X, Sigma , mu ) rightarrow L^1(X, Sigma , mu )) defined on a finite measure space ((X, Sigma , mu )). We give a necessary and sufficient condition for a constrictive Markov operator T which is an integral operator with stochastic kernel satisfying (Tmathbf {1}_X=mathbf {1}_X).