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Proceedings of the American Mathematical Society

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On compact and bounding holomorphic mappings

Author: Mikael Lindström
Journal: Proc. Amer. Math. Soc. 105 (1989), 356-361
MSC: Primary 46G20; Secondary 58C10
MathSciNet review: 933517
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Abstract: Let $ E$ and $ F$ be complex Banach spaces. We say that a holomorphic mapping $ f$ from $ E$ into $ F$ is compact respectively bounding if $ f$ maps some neighbourhood of every point of $ E$ into a relatively compact respectively bounding subset of $ F$. Recall that a subset of $ E$ is bounding if it is mapped onto a bounded set by every complex valued holomorphic mapping on $ E$. Compact holomorphic mappings have been studied by R. Aron and M. Schottenloher in [1]. Since every relatively compact subset of a Banach space is trivially bounding it is clear that every compact holomorphic mapping is bounding. We show that the product of three bounding holomorphic mappings is compact.

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  • [1] R. M. Aron and M. Schottenloher, Compact holomorphic mappings on Banach spaces and the approximation property, J. Funct. Anal. 21 (1976), 7-30. MR 0402504 (53:6323)
  • [2] J. Bourgain and J. Diestel, Limited operators and strict cosingularity, Math. Nachr. 119 (1984), 55-58. MR 774176 (86d:47024)
  • [3] W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński, Factoring weakly compact operators, J. Funct. Anal. 17 (1974), 311-327. MR 0355536 (50:8010)
  • [4] S. Dineen, Complex analysis in locally convex spaces, North Holland 1981. MR 640093 (84b:46050)
  • [5] B. Josefson, Bounding subsets of $ {l^\infty }(A)$, J. Math. Pures et Appl. 57 (1978), 397-421. MR 524627 (81a:46019)
  • [6] M. Lindström, A characterization of Schwartz spaces, Math. Z. 198 (1988), 423-430. MR 946613 (89f:46009)
  • [7] -, Schwartz spaces and compact holomorphic mappings, Manuscripta Math. 60 (1988), 139-144. MR 924083 (89d:46004)
  • [8] A. Pełczyński, A theorem of Dunford-Pettis type for polynomial operators, Bull. Acad. Pol. Sci. 11 (1963), 379-386. MR 0161161 (28:4370)
  • [9] R. A. Ryan, Weakly compact holomorphic mappings on Banach spaces, Pacific J. Math. 131 (1988), 179-190. MR 917872 (89a:46103)

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Article copyright: © Copyright 1989 American Mathematical Society

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