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On isomorphic classification of tensor products $E_{∞}(a) ⊗̂ E'_{∞}(b)$

Seria
Rozprawy Matematyczne tom/nr w serii: 350 wydano: 1996
Zawartość
Warianty tytułu
Abstrakty
EN
Abstract
New linear topological invariants are introduced and utilized to give an isomorphic classification of tensor products of the type $E_{∞}(a) ⊗̂ E'_{∞}(b)$, where $E_{∞}(a)$ is a power series space of infinite type. These invariants are modifications of those suggested earlier by Zahariuta. In particular, some new results are obtained for spaces of infinitely differentiable functions with values in a locally convex space X. These spaces coincide, up to isomorphism, with spaces L(s',X) of all continuous linear operators into X from the dual space of the space s of rapidly decreasing sequences. Most of the results given here with proofs were announced in [12].
EN
CONTENTS
0. Introduction...............................................5
1. Preliminaries.............................................6
2. Power series space-valued case..............8
3. Main results..............................................9
4. F- and DF-subspaces.............................11
5. Quasidiagonal isomorphism....................13
6. Sufficiency...............................................14
7. Linear Topological Invariants (LTI)..........16
8. Necessary conditions..............................20
9. Spaces $s ⊗̂ E'_∞(b)$............................22
10. Spaces $s' ⊗̂ E_{∞}(a)$.......................23
References.................................................26
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 350
Liczba stron
27
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCL
Daty
wydano
1996
otrzymano
1995-03-2
poprawiono
1995-05-09
Twórcy
autor
  • Department of Mathematics, Bilkent University, 06533 Ankara, Turkey
  • State Building Academy, Rostov-on-Don, Russia
autor
  • Department of Mathematics, Marmara Research Center, Gebze-Kocaeli, Turkey
  • Department of Mechanics and Mathematics, Rostov State University, Rostov-on-Don, Russia
Bibliografia
  • [1] A. Aytuna, J. Krone and T. Terzioğlu, Complemented infinite type power series subspaces of nuclear spaces, Math. Ann. 283 (1989), 193-202.
  • [2] V. I. Baran, On quasiequivalence of absolute bases in Cartesian products of Köthe spaces, in: Actual Problems of Mathematical Analysis, Rostov-on-Don, 1978, 13-21 (in Russian).
  • [3] C. Bessaga, Some remarks on Dragilev's theorem, Studia Math. 31 (1968), 307-318.
  • [4] C. Bessaga, A. Pełczyński and S. Rolewicz, On diametral approximative dimension and linear homogeneity of F-spaces, Bull. Acad. Polon. Sci. 9 (1961), 677-683.
  • [5] P. A. Chalov and V. P. Zahariuta, On linear topological invariants, manuscript No. 5941-85, deposited at VINITI, 1985 (in Russian).
  • [6] P. A. Chalov and V. P. Zahariuta, On linear topological invariants on some class of families of Hilbert spaces, manuscript No. 3862-B 86, deposited at VINITI, 1986 (in Russian).
  • [7] M. M. Dragilev, On regular bases in nuclear spaces, Mat. Sb. 68 (1965), 153-173 (in Russian).
  • [8] M. M. Dragilev, On special dimensions, defined on some classes of Köthe spaces, Mat. Sb. 80 (1969), 225-240 (in Russian).
  • [9] M. M. Dragilev, Bases in Köthe spaces, Rostov State Univ., Rostov-on-Don, 1983 (in Russian).
  • [10] E. Dubinsky, The structure of Nuclear Fréchet Spaces, Lecture Notes in Math. 720, Springer, 1979.
  • [11] E. Dubinsky and D. Vogt, Bases in complemented subspaces of power series spaces, Bull. Polish Acad. Sci. 34 (1986), 65-67.
  • [12] A. Goncharov, T. Terzioğlu and V. Zahariuta, On isomorphic classification of spaces $s ⊗̂ E'_∞(a)$, in: Linear Topological Spaces and Complex Analysis, METU-TÜBİTAK, Ankara, 1994, 14-24.
  • [13] A. P. Goncharov and V. P. Zahariuta, Linear topological invariants for tensor products of power F-, DF-spaces, Turkish J. Math. 1 (1995), 90-101.
  • [14] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955).
  • [15] A. N. Kolmogorov, On the linear dimension of topological vector spaces, Dokl. Akad. Nauk SSSR 120 (1958), 239-241 (in Russian).
  • [16] V. P. Kondakov, Questions of geometry of non-normed spaces, Rostov State Univ., Rostov-on-Don, 1983 (in Russian).
  • [17] B. S. Mityagin, Approximative dimension and bases in nuclear spaces, Uspekhi Mat. Nauk 16 (4) (1961), 63-132 (in Russian).
  • [18] B. S. Mityagin, Equivalence of bases in Hilbert scales, Studia Math. 37 (1971), 111-137 (in Russian).
  • [19] B. S. Mityagin, Non-Schwartzian power series spaces, Math. Z. 182 (1983), 303-310.
  • [20] A. Pełczyński, On the approximation of S-spaces by finite-dimensional spaces, Bull. Acad. Polon. Sci. 5 (1957), 879-881.
  • [21] H. H. Schaefer, Topological Vector Spaces, Grad. Texts in Math. 3, Springer, New York, 1971.
  • [22] T. Terzioğlu, Some invariants of Fréchet spaces and imbeddings of smooth sequence spaces, in: Advances in the Theory of Fréchet Spaces, Kluwer, 1989, 305-324.
  • [23] M. Valdivia, Topics in Locally Convex Spaces, North-Holland Math. Stud. 67, North-Holland, Amsterdam, 1982.
  • [24] D. Vogt, Charakterisierung der Unterräume von s, Math. Z. 155 (1977), 109-117.
  • [25] D. Vogt, Eine Charakterisierung der Potenzreihenräume von endlichem Typ und ihre Folgerungen, Manuscripta Math. 37 (1982), 269-301.
  • [26] V. P. Zahariuta, Linear topological invariants and isomorphisms of spaces of analytic functions, Mat. Anal. i Prilozhen., Rostov Univ., 2 (1970), 3-13; 3 (1971), 176-180 (in Russian).
  • [27] V. P. Zahariuta, Some linear topological invariants and isomorphisms of tensor products of scale's centers, Izv. Severo-Kavkaz. Nauchn. Tsentra Vyssh. Shkoly Estestv. Nauk. 4 (1974), 62-64 (in Russian).
  • [28] V. P. Zahariuta, On isomorphisms and quasiequivalence of bases of power Köthe spaces, Dokl. Akad. Nauk SSSR 221 (1975), 772-774 (in Russian).
  • [29] V. P. Zahariuta, On isomorphisms and quasiequivalence of bases of power Köthe spaces, in: Proc. 7th Winter School in Drogobych, 1976, 101-126 (in Russian).
  • [30] V. P. Zahariuta, Generalized Mityagin invariants and continuum pairwise nonisomorphic spaces of analytic functions, Funktsional. Anal. i Prilozhen. 11 (3) (1977), 24-30 (in Russian).
  • [31] V. P. Zahariuta, Synthetic diameters and linear topological invariants, in: School on Theory of Operators in Function Spaces (Abstracts of Reports), Minsk, 1978, 51-52 (in Russian).
  • [32] V. P. Zahariuta, Isomorphism of spaces of vector-valued infinitely differentiable functions, in: School on Theory of Operators in Function Spaces (Abstracts of Reports), Minsk, 1982 (in Russian).
  • [33] V. P. Zahariuta, On isomorphic classification of F-spaces, in: Lecture Notes in Math. 1043, Springer, 1984, 34-37.
  • [34] V. P. Zahariuta, Linear topological invariants and their applications to generalized power spaces, manuscript of survey, Rostov State Univ., 1979 (in Russian); revised English version will appear in Turkish J. Math.
  • [35] V. P. Zahariuta, Linear topological invariants and mixed (F,DF)-spaces, Mathematische Forschungsinstitut Oberwolfach, Tagungsbericht 44/1992, Funktionalanalysis (4.10-10.10.1992), 21.
Języki publikacji
EN
Uwagi
1991 Mathematics Subject Classification: 46A04, 46A45, 46A11, 46A32.
Identyfikator YADDA
bwmeta1.element.zamlynska-6c2c543e-4209-46fe-9868-b7f372fc924b
Identyfikatory
ISSN
0012-3862
Kolekcja
DML-PL
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