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 eigenfun...