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On interpolation by analytic maps in infinite dimensions

Published online by Cambridge University Press:  24 October 2008

J. Globevnik
J. Globevnik
Affiliation:
University of Ljubljana
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Abstract

Let A be the complex Banach algebra of all bounded continuous complex-valued functions on the closed unit ball of a complex Banach space X, analytic on the open unit ball, with sup norm. For a class of spaces X which contains all infinite dimensional complex reflexive spaces we prove the existence of non-compact peak interpolation sets for A. We prove some related interpolation theorems for vector-valued functions and present some applications to the ranges of analytic maps between Banach spaces. We also show that in general peak interpolation sets for A do not exist.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 83 , Issue 2 , March 1978 , pp. 243 - 252
Copyright
Copyright © Cambridge Philosophical Society 1978

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References

REFERENCES

(1)Aron, R. M.The range of vector valued holomorphic mappings. Proc. Conf. on Anal. Funct., Krakow 1974. Ann. Polon. Math. 33 (1976), 17–20.Google Scholar
(2)Day, M. M.Normed linear spaces. Ergebnisse der Math. (Springer, 1973).CrossRefGoogle Scholar
(3)Diestel, J.Geometry of Banach spaces — Selected topics (Lecture Notes in Math. no. 485: Springer, 1975).CrossRefGoogle Scholar
(4)Dunford, N. and Schwartz, J. T.Linear operators. Part I: General theory (Interscience, 1958).Google Scholar
(5)Globevnik, J.The Rudin-Carleson theorem for vector-valued functions. Proc. Amer. Math. Soc. 53 (1975), 250–252.Google Scholar
(6)Globevnik, J.Analytic functions whose range is dense in a ball. J. Funct. Anal. 22 (1976), 32–38.CrossRefGoogle Scholar
(7)Globevnik, J.The range of vector-valued analytic functions. Arkiv för Mat. 14 (1976), 113–118. The range of vector-valued analytic functions: II. Arkiv för Mat. 14 (1976), 297–298.CrossRefGoogle Scholar
(8)Globevnik, J. On the range of analytic functions into a Banach space. To appear in Proc. Symp. Infin. Dim. Holomorphy and Appl., Campinas 1975 (North Holland).Google Scholar
(9)Harris, L. A. Schwarz's lemma and the maximum principle in infinite dimensional spaces. Thesis, Cornell University, 1969.Google Scholar
(10)Hille, E. and Phillips, R. S.Functional analysis and semi-groups. Amer. Math. Soc. Colloq. Publ. no. 31 (1957).Google Scholar
(11)Hoffman, K.Banach spaces of analytic functions (Prentice Hall, 1962).Google Scholar
(12)James, R. C.Characterizations of reflexivity. Studio Math. 23 (1964), 205–216.CrossRefGoogle Scholar
(13)Köthe, G.Topological vector spaces: I. Grundlehr. Math. Wiss. 159 (Springer, 1969).Google Scholar
(14)Leibowitz, G. M.Lectures on complex function algebras (Scott, Foresman, 1970).Google Scholar
(15)Lindenstrauss, J.On operators which attain their norm. Israel J. Math. 1 (1963), 139–148.CrossRefGoogle Scholar
(16)Pełczyński, A.Some linear topological properties of separable function algebras. Proc. Amer. Math. Soc. 18 (1967), 652–661.CrossRefGoogle Scholar
(17)Phelps, R. R.Dentability and extreme points in Banach spaces. J. Funct. Anal. 16 (1974), 78–90.CrossRefGoogle Scholar
(18)Rudin, W.Function theory in polydiscs (Benjamin, 1969).Google Scholar
(19)Rudin, W.Real and complex analysis (McGraw-Hill, 1970).Google Scholar
(20)Rudin, W.Holomorphic maps of discs into F-spaces. Complex Analysis, Kentucky 1976, 104–108. Lecture Notes in Mathematics 599 (Springer, 1977).Google Scholar
(21)Stout, E. L.The theory of uniform algebras (Bogden and Quigley, 1971).Google Scholar
(22)Stout, E. L.On some restriction algebras: Function algebras, ed. Birtel, F. T., pp. 6–11 (Scott, Foresman, 1966).Google Scholar
(23)Thorp, E. and Whitley, R.The strong maximum modulus theorem for analytic functions into a Banach space. Proc. Amer. Math. Soc. 18 (1967), 640–646.CrossRefGoogle Scholar