Completely continuous endomorphisms of p-adic Banach spaces

In Dwork’s memoir [3] concerning the rationality of zeta functions, an essential role is played by the p-adic analytic function det(1− tu), where u is a certain infinite matrix. This analytic function is an entire function, exactly as in the classical Fredholm theory. It was natural to pursue this analogy and extend to u the spectral theory of F. Riesz; this is just what Dwork did ([4], §2). In that which follows, I show that these results simply provide the fact that u is the matrix of a completelycontinuous endomorphism of a Banach space; it is found in fact that, in p-adic analysis, the theories of Riesz and of Fredholm have the same domain of validity: there is no distinction between nuclear operators (or ??) and completely continuous operators.