Recently the Ruelle–Perron–Frobenius theorem was proved for Hölder potentials defined on the symbolic space , where (the alphabet) M is any compact metric space. In this paper, we extend this theorem to the Walters space , in similar general alphabets. We also describe in detail an abstract procedure to obtain the Fréchet 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 Hölder spaces. The analyticity of the pressure functional on Hölder spaces is established. An exponential decay of the correlations is shown when the Ruelle operator has the spectral gap property.