Markov Processes and Semigroups of Operators

In this paper the relations between various semigroups of operators and between various infinitesimal operators connected with a homogeneous in t Markov process are investigated. General conditions are established under which the Markov process is determined by its corresponding infinitesimal operator.
Let $U_t$ be a semigroup of linear operators in the Banach space $L$ such that $\|U_t\|\leq1$. Let $T_t=U_t$ be an adjoint semigroup in the conjugate space $B=L^*$. More abstractly the main object of this paper can be characterized as the study of semigroups $T_t$ and its infinitesimal operators in strong and weak topologies of space $B$.