# Bessaga's conjecture in unstable Köthe spaces and products

Studia Mathematica (1993)

- Volume: 104, Issue: 3, page 221-228
- ISSN: 0039-3223

## Access Full Article

top## Abstract

top## How to cite

topNurlu, Zefer, and Sarsour, Jasser. "Bessaga's conjecture in unstable Köthe spaces and products." Studia Mathematica 104.3 (1993): 221-228. <http://eudml.org/doc/215971>.

@article{Nurlu1993,

abstract = {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.},

author = {Nurlu, Zefer, Sarsour, Jasser},

journal = {Studia Mathematica},

keywords = {complemented subspace of a nuclear Fréchet space; Kolmogorov diameter},

language = {eng},

number = {3},

pages = {221-228},

title = {Bessaga's conjecture in unstable Köthe spaces and products},

url = {http://eudml.org/doc/215971},

volume = {104},

year = {1993},

}

TY - JOUR

AU - Nurlu, Zefer

AU - Sarsour, Jasser

TI - Bessaga's conjecture in unstable Köthe spaces and products

JO - Studia Mathematica

PY - 1993

VL - 104

IS - 3

SP - 221

EP - 228

AB - 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.

LA - eng

KW - complemented subspace of a nuclear Fréchet space; Kolmogorov diameter

UR - http://eudml.org/doc/215971

ER -

## References

top- [1] H. Ahonen, On nuclear spaces defined by Dragilev functions, Ann. Acad. Sci. Fenn. Ser. AI Math. Dissertationes 38 (1981), 1-57. Zbl0483.46009
- [2] C. Bessaga, Some remarks on Dragilev's theorem, Studia Math. 31 (1968), 307-318. Zbl0182.45301
- [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. Zbl0297.46008
- [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. Zbl0403.46005
- [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). Zbl0589.46006
- [11] G. Köthe, Topologische lineare Räume I, Springer, Berlin 1960. Zbl0093.11901
- [12] J. Krone, Bases in the range of operators between Köthe spaces, Doğa Tr. J. Math. 10 (1986), 162-166 (special issue). Zbl0970.46518
- [13] B. S. Mityagin, Approximative dimension and bases in nuclear spaces, Uspekhi Mat. Nauk 16 (4) (1961), 73-132 (in Russian). Zbl0104.08601
- [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. Zbl0612.46009
- [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. Zbl0537.46005
- [18] J. Sarsour, Bessaga's conjecture and quasi-equivalence property in unstable Köthe spaces, Ph.D. Thesis, METU, Ankara 1991. Zbl0812.46004
- [19] T. Terzioğlu, Unstable Köthe spaces and the functor Ext, Doğa Tr. J. Math. 10 (1986), 227-231 (special issue). Zbl0970.46523
- [20] D. Vogt, Eine Charakterisierung der Potenzreihenräume von endlichem Typ und ihre Folgerungen, Manuscripta Math. 37 (1982), 269-301. Zbl0512.46003

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.