At the very end of the paper Formal Cohomology I by Monsky and Washnitzer, they write the following:
"In some sense, the operator $\psi$ applied to a power series gives it "better growth conditions". Now, such operators have been studied by Dwork using p-adic Banach space techniques. In a later paper we shall exploit these methods to obtain a spectral decomposition for the action of $\psi$ on $D(A) \otimes_\mathbb{Z} \mathbb{Q}$, and for the action of $\psi^*$ on $H(\overline{A}; K)$. This spectral decomposition may be thought of as a weak finiteness theorem for the groups $H^i(\overline{A}; K)$".
Using their definitions:
$A$ is a formally smooth weakly complete finitely generated algebra over a complete DVR $(R, \pi)$ of characteristic zero with fraction field $K$ and $|R / \pi R| = q$;
$\overline{A} = A / \pi A$;
$D(A)$ is the module of continuous differentials of $A$ over $R$;
$\phi: A \to A$ is an endomorphism with the reduction $\overline{\phi}: \overline{A} \to \overline{A}$ sending $x \mapsto x^q$. $\phi$ is assumed to be injective and finite;
$S_{A / \phi(A)}: D(A) \to D(A)$ the trace map they define in the paper;
$\psi: D(A) \to D(A)$ sends $\omega \mapsto \phi^{-1}(S_{A / \phi(A)}(\omega))$;
$H(\overline{A}, K)$ is the Monsky-Washnitzer cohomology.
This paper has two sequels (Formal Cohomology II/III), but I looked through both of them and didn't find anything resembling this claimed spectral decomposition. Was this result ever published?
Reference:
Monsky, P., and G. Washnitzer. “Formal Cohomology: I.” Annals of Mathematics, vol. 88, no. 2, 1968, pp. 181–217. JSTOR, https://doi.org/10.2307/1970571.
