Invariant probabilities for discrete time linear dynamics via thermodynamic formalism

We show the existence of invariant ergodic σ-additive probability measures with full support on X for a class of linear operators L : X → X, where L is a weighted shift operator and X either is the Banach space c0(R) or lp(R) for 1 ⩽ p < ∞. In order to do so, we adapt ideas from thermodynamic formalism as follows. For a given bounded Hölder continuous potential A:X→R , we define a transfer operator LA which acts on continuous functions on X and prove that this operator satisfies a Ruelle–Perron–Frobenius theorem. That is, we show the existence of an eigenfunction for LA which provides us with a normalised potential A¯ and an action of the dual operator LA¯* on the one-Wasserstein space of probabilities on X with a unique fixed point, to which we refer to as Gibbs probability. It is worth noting that the definition of LA requires an a priori probability on the kernel of L. These results are extended to a wide class of operators with a non-trivial kernel defined on separable Banach spaces.