  Denote by $X$ a Banach space and by $T : X \to X$ a bounded linear operator
with non-trivial kernel satisfying suitable conditions. We consider the
concepts of entropy - for $T$-invariant probability measures - and pressure for
H\"older continuous potentials. We also prove the existence of ground states
(the limit when temperature goes to zero) associated with such class of
potentials when the Banach space $X$ is equipped with a Schauder basis. We
produce an example concerning weighted shift operators defined on the Banach
spaces $c_0(\mathbb{R})$ and $l^p(\mathbb{R})$, $1 \leq p < +\infty$, where our
results do apply. In addition, we prove the existence of calibrated sub-actions
when the potential satisfies certain regularity conditions using properties of
the so-called Ma\~n\'e potential. We also exhibit examples of selection at zero
temperature and explicit sub-actions in the class of H\"older continuous
potentials.
