  Transfer operators and Ruelle zeta functions for super-continuous functions
on one-sided topological Markov shifts are considered. For every
super-continuous function, we construct a Banach space on which the associated
transfer operator is compact. Using this Banach space, we establish the trace
formula and spectral representation of Ruelle zeta functions for a certain
class of super-continuous functions. Our results include, as a special case,
the classical trace formula and spectral representation for the class of
locally constant functions.
