The ergodic theorems for Markov chains with an arbitrary phase space

The ergodic theorems are formulated both is the case of the uniform ergodicity and in the absence of this latter under more general conditions than before. The conditions laid down on the transition function in the case of nonuniform ergodicity are less restrictive than these by Athreya-Ney and Nummelin. In contrast to these latter the restrictions are set not on the initial transition function, but on that of the imbedded Markov chain which is constructed by the instants of the recurrence to some fixed set. The proof is based on the spectral theory of Banach algebras.