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

Manfred Scheve

ArticleOriginal scientific text

Title

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

Authors ¹

Affiliations

Mathematisches Institut, Universität Düsseldorf, Có Prof. R. Meise, Universitätsstraße 1, D-4000 Düsseldorf, Germany

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.

P. J. Boland and S. Dineen, Holomorphic functions on fully nuclear spaces, Bull. Soc. Math. France 106 (1978), 311-336.
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).
P. B. Djakov, A short proof of the theorem of Crone and Robinson on quasi-equivalence of regular bases, Studia Math. 53 (1975), 269-271.
H. Jarchow, Locally Convex Spaces, Teubner, 1981.
R. Meise and D. Vogt, Structure of spaces of holomorphic functions on infinite dimensional polydiscs, Studia Math. 75 (1983), 235-252.
R. Meise and D. Vogt, Analytic isomorphisms of infinite dimensional polydiscs and an application, Bull. Soc. Math. France 111 (1983), 3-20.
R. Meise and D. Vogt, Holomorphic functions of uniformly bounded type on nuclear Fréchet spaces, Studia Math. 83 (1986), 147-166.
R. Meise and D. Vogt, Holomorphic Functions on Nuclear Sequence Spaces, Departamento de Teoría de Funciones, Universidad Complutense, Madrid 1986.
L. Mirsky, Transversal Theory, Academic Press, 1971.
B. S. Mityagin, The equivalence of bases in Hilbert scales, Studia Math. 37 (1971), 111-137 (in Russian).
A. Pietsch, Nuclear Locally Convex Spaces, Ergeb. Math. Grenzgeb. 66, Springer, 1972.
H. H. Schaefer, Topological Vector Spaces, Springer, 1971.
M. Scheve, Räume holomorpher Funktionen auf unendlich-dimensionalen Polyzylindern, Dissertation, Düsseldorf 1988.
D. Vogt, Frécheträume, zwischen denen jede stetige lineare Abbildung beschränkt ist, J. Reine Angew. Math. 345 (1983), 182-200.
M. J. Wagner, Unterräume und Quotienten von Potenzreihenräumen, Dissertation, Wuppertal 1977.
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).

Title

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

Affiliations

Abstract

Bibliography