The double transpose of the Ruelle operator

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