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Weak-Star Continuous Analytic Functions

Published online by Cambridge University Press:  20 November 2018

R. M. Aron ,
B. J. Cole  and
T. W. Gamelin
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Abstract

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Let 𝒳 be a complex Banach space, with open unit ball B. We consider the algebra of analytic functions on B that are weakly continuous and that are uniformly continuous with respect to the norm. We show these are precisely the analytic functions on B that extend to be weak-star continuous on the closed unit ball of 𝒳**. If 𝒳* has the approximation property, then any such function is approximable uniformly on B by finite polynomials in elements of 𝒳*. On the other hand, there exist Banach spaces for which these finite-type polynomials fail to approximate. We consider also the approximation of entire functions by finite-type polynomials. Assuming 𝒳* has the approximation property, we show that entire functions are approximable uniformly on bounded sets if and only if the spectrum of the algebra of entire functions coincides (as a point set) with 𝒳**.

Keywords

46G2046E1546J1546J20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 4 , 01 August 1995 , pp. 673 - 683
Copyright
Copyright © Canadian Mathematical Society 1995

References

1. Aron, R.M. and Berner, P.D., A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France 106(1978), 3–24.Google Scholar
2. Aron, R.M., Cole, B.J. and Gamelin, T.W., Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math. 415(1991), 51–93.Google Scholar
3. Aron, R.M., Hervés, C. and Valdivia, M., Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52(1983), 189–204.Google Scholar
4. Carne, T.K., Cole, B.J. and Gamelin, T.W., A uniform algebra of analytic functions on a Banach space, Trans. Amer. Math. Soc. 314(1989), 639–659.Google Scholar
5. Davie, A.M. and Gamelin, T.W., A theorem on polynomial-star approximation, Proc. Amer. Math. Soc. 106(1989), 351–356.Google Scholar
6. Dunford, N. and Schwartz, J., Linear Operators, Part I, Interscience, 1959.Google Scholar
7. Gamelin, T.W., Uniform Algebras, Second Edition, Chelsea Press, New York, 1984.Google Scholar
8. Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces, Vol. 1, Springer-Verlag, Berlin, 1977.Google Scholar
9. Moraes, L., Extension ofholomorphic mappings from E to E”, Proc. Amer. Math. Soc. 118(1993), 455–461.Google Scholar
10. Mujica, J., Complex Analysis in Banach Spaces, Math. Studies 120, North Holland, 1986.Google Scholar
11. Nachbin, L., Topology on Spaces of Holomorphic Mappings, Springer-Verlag, Berlin, 1969.Google Scholar
12. Pelczynski, A., A property of multilinear operations, Studia Math. 16(1957), 173–182.Google Scholar
13. Tsirelson, B.S., Not every Banach space contains an imbedding of ℓp or CQ, Functional Anal. Appl. 8(1974), 138–141.Google Scholar