Construction of standard exact sequences of power series spaces

• Autorzy: Markus Poppenberg , Dietmar Vogt
• Języki publikacji: EN

Abstrakty

The following result is proved: Let $Λ_R^p(α)$ denote a power series space of infinite or of finite type, and equip $Λ_R^p(α)$ with its canonical fundamental system of norms, R ∈ {0,∞}, 1 ≤ p < ∞. Then a tamely exact sequence (⁎) $0 → Λ_{R}^{p}(α) → Λ_{R}^{p}(α) → Λ_{R}^{p}(α)^ℕ → 0$ exists iff α is strongly stable, i.e. $lim_n α_{2n}/α_n = 1$, and a linear-tamely exact sequence (*) exists iff α is uniformly stable, i.e. there is A such that $lim sup_n α_{Kn}/α_n ≤ A < ∞$ for all K. This result extends a theorem of Vogt and Wagner which states that a topologically exact sequence (*) exists iff α is stable, i.e. $sup_n α_{2n}/α_n < ∞$.

Kategorie tematyczne

• 46A45: Sequence spaces (including K\"othe sequence spaces)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1994-1995
• Tom: 112
• Numer: 3
• Strony: 229-241

Daty

wydano

1995

otrzymano

1993-09-13

poprawiono

1994-05-05

Twórcy

autor

Markus Poppenberg

• Fachbereich Mathematik, Universität Dortmund, Vogelpothsweg 87, D-44221 Dortmund, Germany

autor

Dietmar Vogt

• Fachbereich Mathematik, Bergische Universität—GH Wuppertal, Gauss-Str. 20, D-42097 Wuppertal, Germany

Bibliografia

• [1] R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. 7 (1982), 65-222.
• [2] H. Külkens, Gleichmäßig shiftstabile Potenzreihenräume, Diplomarbeit, Wuppertal, 1991.
• [3] A. Pietsch, Nuclear Locally Convex Spaces, Ergeb. Math. Grenzgeb. 66, Springer, 1972.
• [4] M. Poppenberg and D. Vogt, A tame splitting theorem for exact sequences of Fréchet spaces, Math. Z., to appear.
• [5] M. Poppenberg and D. Vogt, Tame splitting theory for Fréchet-Hilbert spaces, in: Functional Analysis, K. D. Bierstedt, A. Pietsch, W. M. Ruess and D. Vogt (eds.), Lecture Notes in Pure and Appl. Math. 150, Marcel Dekker, New York, 1994, 475-492.
• [6] D. Vogt, On the characterization of subspaces and quotient spaces of stable power series spaces of finite type, Arch. Math. (Basel) 50 (1988), 463-469.
• [7] D. Vogt, Eine Charakterisierung der Potenzreihenräume von endlichem Typ und ihre Folgerungen, Manuscripta Math. 37 (1982), 269-301.
• [8] D. Vogt, Charakterisierung der Unterräume eines nuklearen stabilen Potenzreihenraumes von endlichem Typ, Studia Math. 71 (1982), 251-270.
• [9] D. Vogt, Tame spaces and power series spaces, Math. Z. 196 (1987), 523-536.
• [10] D. Vogt, Power series space representations of nuclear Fréchet spaces, Trans. Amer. Math. Soc. 319 (1990), 191-208.
• [11] D. Vogt and M. J. Wagner, Charakterisierung der Unterräume und Quotientenräume der nuklearen stabilen Potenzreihenräume von unendlichem Typ, Studia Math. 70 (1981), 63-80.