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 $\|U_t \| \leqq 1$. 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.