Isomorphy classes of spaces of holomorphic functions on open polydiscs in dual power series spaces

Manfred Scheve

Studia Mathematica (1991)

  • Volume: 101, Issue: 1, page 83-104
  • ISSN: 0039-3223

Abstract

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Let Λ_R(α) be a nuclear power series space of finite or infinite type with lim_{j→∞} (1/j) log α_j = 0. We consider open polydiscs D_a in Λ_R(α)'_b with finite radii and the spaces H(D_a) of all holomorphic functions on D_a under the compact-open topology. We characterize all isomorphy classes of the spaces {H(D_a) | a ∈ Λ_R(α), a > 0}. In the case of a nuclear power series space Λ₁(α) of finite type we give this characterization in terms of the invariants (Ω̅ ) and (Ω̃ ) known from the theory of linear operators between Fréchet spaces.

How to cite

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Scheve, Manfred. "Isomorphy classes of spaces of holomorphic functions on open polydiscs in dual power series spaces." Studia Mathematica 101.1 (1991): 83-104. <http://eudml.org/doc/215894>.

@article{Scheve1991,
abstract = {Let Λ\_R(α) be a nuclear power series space of finite or infinite type with lim\_\{j→∞\} (1/j) log α\_j = 0. We consider open polydiscs D\_a in Λ\_R(α)'\_b with finite radii and the spaces H(D\_a) of all holomorphic functions on D\_a under the compact-open topology. We characterize all isomorphy classes of the spaces \{H(D\_a) | a ∈ Λ\_R(α), a > 0\}. In the case of a nuclear power series space Λ₁(α) of finite type we give this characterization in terms of the invariants (Ω̅ ) and (Ω̃ ) known from the theory of linear operators between Fréchet spaces.},
author = {Scheve, Manfred},
journal = {Studia Mathematica},
keywords = {nuclear power series space of finite or infinite type; holomorphic functions; compact-open topology; linear operators between Fréchet spaces},
language = {eng},
number = {1},
pages = {83-104},
title = {Isomorphy classes of spaces of holomorphic functions on open polydiscs in dual power series spaces},
url = {http://eudml.org/doc/215894},
volume = {101},
year = {1991},
}

TY - JOUR
AU - Scheve, Manfred
TI - Isomorphy classes of spaces of holomorphic functions on open polydiscs in dual power series spaces
JO - Studia Mathematica
PY - 1991
VL - 101
IS - 1
SP - 83
EP - 104
AB - Let Λ_R(α) be a nuclear power series space of finite or infinite type with lim_{j→∞} (1/j) log α_j = 0. We consider open polydiscs D_a in Λ_R(α)'_b with finite radii and the spaces H(D_a) of all holomorphic functions on D_a under the compact-open topology. We characterize all isomorphy classes of the spaces {H(D_a) | a ∈ Λ_R(α), a > 0}. In the case of a nuclear power series space Λ₁(α) of finite type we give this characterization in terms of the invariants (Ω̅ ) and (Ω̃ ) known from the theory of linear operators between Fréchet spaces.
LA - eng
KW - nuclear power series space of finite or infinite type; holomorphic functions; compact-open topology; linear operators between Fréchet spaces
UR - http://eudml.org/doc/215894
ER -

References

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  1. [1] P. J. Boland and S. Dineen, Holomorphic functions on fully nuclear spaces, Bull. Soc. Math. France 106 (1978), 311-336. Zbl0402.46017
  2. [2] P. A. Chalov and V. P. Zakharyuta, A quasiequivalence criterion for absolute bases in an arbitrary (F)-space, Izv. Severo-Kavkaz. Nauchn. Tsentra Vyssh. Shkoly Estestv. Nauki 1983 (2), 22-24 (in Russian). Zbl0533.46003
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  8. [8] R. Meise and D. Vogt, Holomorphic Functions on Nuclear Sequence Spaces, Departamento de Teoría de Funciones, Universidad Complutense, Madrid 1986. Zbl0657.46003
  9. [9] L. Mirsky, Transversal Theory, Academic Press, 1971. 
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  11. [11] A. Pietsch, Nuclear Locally Convex Spaces, Ergeb. Math. Grenzgeb. 66, Springer, 1972. 
  12. [12] H. H. Schaefer, Topological Vector Spaces, Springer, 1971. 
  13. [13] M. Scheve, Räume holomorpher Funktionen auf unendlich-dimensionalen Polyzylindern, Dissertation, Düsseldorf 1988. 
  14. [14] D. Vogt, Frécheträume, zwischen denen jede stetige lineare Abbildung beschränkt ist, J. Reine Angew. Math. 345 (1983), 182-200. 
  15. [15] M. J. Wagner, Unterräume und Quotienten von Potenzreihenräumen, Dissertation, Wuppertal 1977. 
  16. [16] V. P. Zakharyuta, Isomorphism and quasiequivalence of bases for Köthe power series spaces, in: Mathematical Programming and Related Problems (Proc. 7th Winter School, Drogobych 1974), Theory of Operators in Linear Spaces, Akad. Nauk SSSR, Tsentr. Ekon.-Mat. Inst., Moscow 1976, 101-126 (in Russian); see also Dokl. Akad. Nauk SSSR 221 (1975), 772-774 (in Russian). 

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