12 - Some remarks on the compact non-nuclear operator problem

Summary

Let us consider the following long standing question:

(*) Suppose that X, Y are infinite dimensional Banach spaces. Does then exist a compact non-nuclear operator T:X → Y?

The problem has positive answer under quite general assumptions. Positive answer of the finite–dimensional version of (*) was given in. Close to (*) is another problem posed by Grothendieck and solved by the example of Pisier–s space P: P is an infinite dimensional Banach space such that

1. 1) p ⊗ _εp = p ⊗ _πp

and

1. 2) P and P' are of cotype 2.

G. Pisier already observed that if P were reflexive or merely if P^* had R.N. property then every bounded operator T:P → P' would be nuclear and thus (*) would have negative answer. Indeed, 1) implies that every T:P → P' is integral. Unfortunately no reflexive Pisier's space P is known to exist. We may also translate the problem into the category of locally convex spaces:

Suppose that X, Y are non-nuclear locally convex spaces. Does then exist a compact non-nuclear operator T:X→Y? This generalized problem has a negative answer,: There is a non-nuclear F space X and a non-nuclear DF space Z = Y' such that every continuous operator T: X→Z is (strongly) nuclear. X and Z are obtained via Pisier's observation mentioned above from an example with properties similar to 1):

There are non-nuclear F spaces X, Y such that X ⊗ _ε Y = X ⊗ _ε Y. Moreover X, Y are hilbertizable.