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Spaces of operators between Fréchet spaces. (English) Zbl 0804.46011

Let \(E\) and \(F\) be Fréchet spaces. A (continuous linear) operator \(T\in L(E,F)\) is said to be Montel, denoted \(T\in M(E,F)\), if it maps bounded subsets of \(E\) into relatively compact subsets of \(F\). In this paper the authors give conditions for pairs of Fréchet spaces implying that \(L(E,F)= M(E,F)\) if and only if \(E\) or \(F\) is Montel. They use as a tool the extensions of the Josefson-Nissenzweig theorem to Fréchet spaces they have recently obtained in colaboration with T. Schlumprecht and M. Valdivia.
As a consequence they prove that a space \(F\) contains a copy of \(c_ 0\) if and only if \(M(E,F)\neq L(E,F)\) (or \(M(E,F)\) is uncomplemented in \(L(E,F)\)) for every space \(E\) which is not Montel. Analogously they prove that a space \(E\) contains a complemented copy of \(\ell_ 1\) if and only if \(M(E,F)\neq L(E,F)\) (or \(M(E,F)\) is uncomplemented in \(L(E,F)\)) for every space \(F\) which is not Montel. Moreover, they characterize the pairs \((E,F)\) of Köthe echelon spaces such that \(M(E,F)\) is complemented in \(L(E,F)\).

MSC:

46A32 Spaces of linear operators; topological tensor products; approximation properties
46A04 Locally convex Fréchet spaces and (DF)-spaces
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46A45 Sequence spaces (including Köthe sequence spaces)
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