Invariant probabilities for discrete time Linear Dynamics via Thermodynamic Formalism

We show the existence of invariant ergodic $\sigma$-additive probability measures with full support on $X$ for a class of linear operators $L:X\to X$, where $L$ is a weighted shift operator and $X$ either is the Banach space $c_0(\mathbb{R})$ or $l^p(\mathbb{R})$ for $1\le p<\mathrm{\infty }$. In order to do so, we adapt ideas from Thermodynamic Formalism as follows. For a given bounded H"older continuous potential $A:X\to \mathbb{R}$, we define a transfer operator ${\mathcal{L}}_A$ 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 ${\mathcal{L}}_A$ which provides us with a normalized potential $\bar{A}$ and an action of the dual operator ${\mathcal{L}}_{\bar{A}}^{\ast }$ on the $1$-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 ${\mathcal{L}}_A$ requires an {\it 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.