In the celebrating paper "Completely continuous endomorphisms of p-adic Banach spaces", Serre established a Fredholm-Riesz theory for compact endomorphisms of Banach spaces over (spherically complete) non-Archimedean field.

Later, people have two different generalizations. In one direction, people fix the ring and generalize the vector spaces , i.e people establish the Fredholm-Riesz theory for nuclear endomorphisms of locally convex vector spaces. On the other direction, people establish the Fredholm-Riesz theory for compact operators of (ON-able) Banach modules over general (Noetherian) Banach rings.

Did there exist a combination of these two generalization?Say, a Fredholm-Riesz theory for "Nuclear endomorphisms" of certain "locally convex" modules over general Banach ring? If this kind of generalization does not exist in general, what's the essential obstruction?


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