Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Tameness of pairs of nuclear power series spaces and related topics

Author: Kaisa Nyberg
Journal: Trans. Amer. Math. Soc. 283 (1984), 645-660
MSC: Primary 46A45; Secondary 46A12
MathSciNet review: 737890
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The equivalence of the following six assertions is proved: (i) The set of the finite limit points of the ratios $ {\alpha _m}/{\beta _n},n,m \in {\mathbf{N}}$, is bounded, (ii) Every operator from $ {\Lambda _\infty }(\beta )$ to $ {\Lambda _1}(\alpha )$ is compact, (iii) The pair $ ({\Lambda _\infty }(\beta ),\,{\Lambda _1}(\alpha ))$ is tame, i.e., for every operator $ T$ from $ {\Lambda _\infty }(\beta )$ to $ {\Lambda _1}(\alpha )$ there is a positive integer $ a$ such that for every $ k \in {\mathbf{N}}$ there is a constant $ {C_k}$ such that $ \vert\vert Tx\vert{\vert _k} \leqslant {C_k}\vert x{\vert _{ak}}$ for every $ x \in {\Lambda _\infty }(\beta )$. (iv) Every short exact sequence $ 0 \to {\Lambda _\tau }(\beta ) \to X \to {\Lambda _1}(\alpha ) \to 0$, where $ \tau = 1$ or $ \infty $, splits. (v) $ {\Lambda _1}(\alpha ) \times {\Lambda _\infty }(\beta )$ has a regular basis, (vi) $ {\Lambda _1}(\alpha ) \otimes {\Lambda _\infty }(\beta )$ has a regular basis. Also the finite type power series spaces that contain subspaces isomorphic to an infinite type power series space are characterized as well as the infinite type power series spaces that have finite type quotient spaces.

References [Enhancements On Off] (What's this?)

  • [1] L. Crone, Ed Dubinsky, and W. B. Robinson, Regular bases in products of power series spaces, J. Functional Analysis 24 (1977), no. 3, 211–222. MR 0435782
  • [2] N. De Grande-De Kimpe and W. B. Robinson, Compact maps and embeddings from an infinite type power series space to a finite type power series space, J. Reine Angew. Math. 293/294 (1977), 52–61. MR 0442639
  • [3] M. M. Dragilev, Riesz classes and multiple regular bases, Teor. Funkciĭ Funkcional. Anal. i Priložen. 15 (1972), 55–78 (Russian). MR 0306858
  • [4] Ed Dubinsky, Infinite type power series subspaces of finite type power series spaces, Israel J. Math. 15 (1973), 257–281. MR 0346483,
  • [5] Ed Dubinsky, The structure of nuclear Fréchet spaces, Lecture Notes in Mathematics, vol. 720, Springer, Berlin, 1979. MR 537039
  • [6] V. V. Kaširin, Subspaces of a finite center of an absolute Riesz scale that are isomorphic to an infinite center, Sibirsk. Mat. Ž. 16 (1975), no. 4, 863–865, 887 (Russian). MR 0388041
  • [7] -, Isomorphic embeddings to some generalized power series spaces, preprint, Warsaw, 1979.
  • [8] Vasilii V. Kashirin, Comparison of linear dimensions of some generalized power series spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 7-8, 567–569 (1980) (English, with Russian summary). MR 581553
  • [9] V. V. Kashirin, Isomorphic embeddings of some generalized power series spaces, Studia Math. 71 (1981/82), no. 2, 169–178. MR 654672
  • [10] Timo Ketonen and Kaisa Nyberg, Twisted sums of nuclear Fréchet spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), no. 2, 323–335. MR 686648,
  • [11] B. S. Mitjagin, Equivalence of bases in Hilbert scales, Studia Math. 37 (1970/71), 111–137 (Russian). MR 0322470
  • [12] B. S. Mitjagin and G. M. Henkin, Linear problems of complex analysis, Uspehi Mat. Nauk 26 (1971), no. 4 (160), 93–152 (Russian). MR 0287297
  • [13] Z. Nurlu, On basic sequences in some Köthe spaces and existence of non-compact operators, Thesis, Potsdam, 1981.
  • [14] -, On pairs of Köthe spaces between which all operators are compact (preprint).
  • [15] Dietmar Vogt and Max-Josef Wagner, Charakterisierung der Unterräume und Quotientenräume der nuklearen stabilen Potenzreihenräume von unendlichem Typ, Studia Math. 70 (1981), no. 1, 63–80 (German, with English summary). MR 646960
  • [16] Dietmar Vogt, Eine Charakterisierung der Potenzreihenräume von endlichem Typ und ihre Folgerungen, Manuscripta Math. 37 (1982), no. 3, 269–301 (German, with English summary). MR 657522,
  • [17] -, Frécheträume zwischendenen jede stetige lineare Abbildung beschränkt ist, J. Reine Angew. Math. (to appear).
  • [18] V. P. Zahariuta, On the isomorphism of cartesian products of locally convex spaces, Studia Math. 46 (1973), 201–221. MR 0330991
  • [19] -, On the isomorphism and quasiequivalence of bases in Köthe sequence spaces, Seventh Winter School Math. Programming, (Drogobych, 1974), 101-126. (Russian)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46A45, 46A12

Retrieve articles in all journals with MSC: 46A45, 46A12

Additional Information

Article copyright: © Copyright 1984 American Mathematical Society