The Double Transpose of the Ruelle Operator

In this paper we study the double transpose of the $L^1(X,\mathscr{B}(X),\nu)$-extensions of the Ruelle transfer operator $\mathscr{L}_{f}$ associated to a general real continuous potential $f\in C(X)$, where $X=E^{\mathbb{N}}$, the alphabet $E$ is any compact metric space and $\nu$ is a maximal eigenmeasure. For this operator, denoted by $\mathbb{L}^{**}_{f}$, we prove the existence of some non-negative eigenfunction, in the Banach lattice sense, associated to $\rho(\mathscr{L}_{f})$, the spectral radius of the Ruelle operator acting on $C(X)$. As an application, we obtain a sufficient condition ensuring that the natural extension of the Ruelle operator to $L^1(X,\mathscr{B}(X),\nu)$ has an eigenfunction associated to $\rho(\mathscr{L}_{f})$. These eigenfunctions agree with the usual maximal eigenfunctions, when the potential $f$ belongs to the H\"older, Walters or Bowen class. We also construct solutions to the classical and generalized variational problem, using the eigenvector constructed here.