The spectral method and Markov chains with an arbitrary phase space

The modification of the spectral method is elaborated, which is appropriate for proving limit theorems for the general Markov chains in a sense of ergodic properties. The essence of this method consists in applying to Markov chains the spectral theory of Banach algebras. The first version of the spectral method was suggested by the author of the present talk in 1957. Since then this method was used in many papers of different authors. However, until the present time the application of the spectral method was restricted to the uniformly ergodic Markov chains, i.e. the chains for which the rate of convergence to the stationary distribution does not depend on an initial state. However, many of Markov chains which are used in applications, in particular denumerable Markov chains, are not uniformly ergodic. The progress is achieved owing to hitherto unknown representation of a resolvent operator in a functional space.