  The theory of $L^2$-spectral gaps for reversible Markov chains has been
studied by many authors. In this paper we consider positive recurrent general
state space Markov chains with stationary transition probabilities. Replacing
the assumption of reversibility by a less strong one, we still obtain a simple
necessary and sufficient condition for the spectral gap property of the
associated Markov operator in terms of isoperimetric constant. Moreover, we
define a new sequence of isoperimetric constants which provides a necessary and
sufficient condition for the existence of a spectral gap in a very general
setting. Finally, these results are used to obtain simple sufficient conditions
for the existence of a spectral gap in terms of the first and second order
transition probabilities.
