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

Spectral Properties of the Ruelle Operator on the Walters Class over Compact Spaces