A Floquet-Liapunov theorem in Fréchet spaces

George N. Galanis; Efstathios E. Vassiliou

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We will show that for each sequence of quasinormable Fréchet spaces ${(E_{n})}_{ℕ}$ there is a Köthe space λ such that $E x t^{1} (λ (A), λ (A) = E x t^{1} (λ (A), E_{n}) = 0$ and there are exact sequences of the form $. . . \to λ (A) \to λ (A) \to λ (A) \to λ (A) \to E_{n} \to 0$ . If, for a fixed ℕ, $E_{n}$ is nuclear or a Köthe sequence space, the resolution above may be reduced to a short exact sequence of the form $0 \to λ (A) \to λ (A) \to E_{n} \to 0$ . The result has some applications in the theory of the functor $E x t^{1}$ in various categories of Fréchet spaces by providing a substitute for non-existing projective resolutions.