  Recently the Ruelle-Perron-Fr\"obenius theorem was proved for H\"older
potentials defined on the symbolic space $\Omega=M^{\mathbb{N}}$, where (the
alphabet) $M$ is any compact metric space. In this paper, we extend this
theorem to the Walters space $W(\Omega)$, in similar general alphabets. We also
describe in detail an abstract procedure to obtain the Fr\'echet-analyticity of
the Ruelle operator under quite general conditions and we apply this result to
prove the analytic dependence of this operator on both Walters and H\"older
spaces. The analyticity of the pressure functional on H\"older spaces is
established. An exponential decay of the correlations is shown when the Ruelle
operator has the spectral gap property.
  A new (and natural) family of Walters potentials (on a finite alphabet
derived from the Ising model) not having an exponential decay of the
correlations is presented. Because of the lack of exponential decay, for such
potentials we have the absence of the spectral gap for the Ruelle operator. The
key idea to prove the lack of exponential decay of the correlations are the
Griffiths-Kelly-Sherman inequalities.
