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1993 | 104 | 3 | 221-228
Tytuł artykułu

Bessaga's conjecture in unstable Köthe spaces and products

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Let F be a complemented subspace of a nuclear Fréchet space E. If E and F both have (absolute) bases $(e_n)$ resp. $(f_n)$, then Bessaga conjectured (see [2] and for a more general form, also [8]) that there exists an isomorphism of F into E mapping $f_n$ to $t_n e_{π(k_n)}$ where $(t_n)$ is a scalar sequence, π is a permutation of ℕ and $(k_n)$ is a subsequence of ℕ. We prove that the conjecture holds if E is unstable, i.e. for some base of decreasing zero-neighborhoods $(U_n)$ consisting of absolutely convex sets one has ∃s ∀p ∃q ∀r $lim_n (d_{n+1}(U_q, U_p))/(d_n(U_r, U_s)) = 0$ where $d_n(U,V)$ denotes the nth Kolmogorov diameter.
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  • Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey
  • Department of Mathematics, The Islamic University of Gaza, P.O. Box 108, Gaza, Gaza Strip
Bibliografia
  • [1] H. Ahonen, On nuclear spaces defined by Dragilev functions, Ann. Acad. Sci. Fenn. Ser. AI Math. Dissertationes 38 (1981), 1-57.
  • [2] C. Bessaga, Some remarks on Dragilev's theorem, Studia Math. 31 (1968), 307-318.
  • [3] L. Crone and W. B. Robinson, Every nuclear Fréchet space with a regular basis has the quasi-equivalence property, ibid. 52 (1975), 203-207.
  • [4] M. M. Dragilev, On special dimensions defined on some classes of Köthe spaces, Math. USSR-Sb. 9 (2) (1968), 213-228.
  • [5] M. M. Dragilev, On regular bases in nuclear spaces, Amer. Math. Soc. Transl. (2) 93 (1970), 61-82.
  • [6] E. Dubinsky, The Structure of Nuclear Fréchet Spaces, Lecture Notes in Math. 720, Springer, Berlin 1979.
  • [7] M. Hall, Jr., Combinatorial Theory, Blaisdell-Waltham, 1967.
  • [8] V. P. Kondakov, On a certain generalization of power series spaces, in: Current Problems in Mathematical Analysis, Rostov State Univ., 1978, 92-99 (in Russian).
  • [9] V. P. Kondakov, Properties of bases of some Köthe spaces and their subspaces, in: Functional Analysis and its Applications 14, Rostov State Univ., 1980, 58-59 (in Russian).
  • [10] V. P. Kondakov, Unconditional bases in certain Köthe spaces, Sibirsk. Mat. Zh. 25 (3) (1984), 109-119 (in Russian).
  • [11] G. Köthe, Topologische lineare Räume I, Springer, Berlin 1960.
  • [12] J. Krone, Bases in the range of operators between Köthe spaces, Doğa Tr. J. Math. 10 (1986), 162-166 (special issue).
  • [13] B. S. Mityagin, Approximative dimension and bases in nuclear spaces, Uspekhi Mat. Nauk 16 (4) (1961), 73-132 (in Russian).
  • [14] B. S. Mityagin, Equivalence of bases in Hilbert scales, Studia Math. 37 (1971), 111-137 (in Russian).
  • [15] Z. Nurlu, On pairs of Köthe spaces between which all operators are compact, Math. Nachr. 122 (1985), 272-287.
  • [16] A. Pietsch, Nuclear Locally Convex Spaces, Springer, Berlin 1972.
  • [17] J. Prada, On idempotent operators on Fréchet spaces, Arch. Math. (Basel) 43 (1984), 179-182.
  • [18] J. Sarsour, Bessaga's conjecture and quasi-equivalence property in unstable Köthe spaces, Ph.D. Thesis, METU, Ankara 1991.
  • [19] T. Terzioğlu, Unstable Köthe spaces and the functor Ext, Doğa Tr. J. Math. 10 (1986), 227-231 (special issue).
  • [20] D. Vogt, Eine Charakterisierung der Potenzreihenräume von endlichem Typ und ihre Folgerungen, Manuscripta Math. 37 (1982), 269-301.
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