Ratio limit theorems for self-adjoint operators and symmetric Markov chains

A simplest ratio limit theorem is obtained for self-adjoint operators in the spaces of L-2 type which leave invariant a cone of nonnegative elements. By means of the theorem we establish ratio limit theorems for symmetric Markov chains and symmetric kernels in measurable spaces. In particular, it is shown that fur symmetric Harris recurrent Markov chains a result is valid which is an analogue of the known Grey theorem (1961) about discrete recurrent symmetric chains. Similar statements are valid for nonnegative symmetric quasi-Feller kernels on locally compact spaces which are Liouville in a certain sense.