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We show the existence of invariant ergodic $\sigma$-additive probabilitymeasures 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 $1\leq p<\infty$. In order to do so,we adapt ideas from Thermodynamic Formalism as follows. For a given boundedH\"older continuous potential $A:X \to \mathbb{R}$, we define a transferoperator $\mathcal{L}_A$ which acts on continuous functions on $X$ and provethat this operator satisfies a Ruelle-Perron-Frobenius theorem. That is, weshow the existence of an eigenfunction for $\mathcal{L}_A$ which provides uswith a normalized potential $\overline{A}$ and an action of the dual operator$\mathcal{L}_{\overline{A}}^*$ 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 apriori} probability on the kernel of $L$. These results are extended to a wideclass of operators with a non-trivial kernel defined on separable Banachspaces.

Invariant probabilities for discrete time Linear Dynamics via Thermodynamic Formalism