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- A survey of some results and open problems in weighted inductive limits and projective description for spaces of holomorphic functions
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A survey of some results and open problems in weighted inductive limits and projective description for spaces of holomorphic functions
Abstract
The development in countable inductive limits of weighted Banach spaces of continuous functions and their projective description had been surveyed before. This case can nowadays be considered as settled completely ; all important properties are characterized, even for (LF)-spaces. The present survey, however, is devoted to a description of the “state of the art” in countable inductive limits H (G) and oH(G) of weighted Banach spaces of holomorphic functions and their projective description. Here many problems have remained open. It is known for quite a while that H(G) = H(G) holds topologically whenever the inductive limit H(G) is a (semi-) Montel space. Counterexamples to projective description for O-growth conditions were given only during the last years, by Bonet and his co-authors Taskinen, Melikhov, Vogt, respectively. But it is not known if there are any counterexamples for oH(G) ; i.e., in the case of o-growth conditions. Among other things, we discuss the analogy between Köthe sequence spaces and spaces of holomorphic functions, the use of the predual of H(G), and the biduality of oH(G) and H(G). One section is devoted to vector valued projective description in a good situation and its connection to an operator representation. Interesting new problems arise here. Finally, the article concludes with some remarks on the proper (Lf)-case, in which, despite various efforts, no general positive projective description result is known for spaces of holomorphic functions.
weigted inductive limits of spaces of continuous or holomorphic functions, projective description, Köthe echelon spaces, predual, biduality, approximation property, counterexamples, vector valued projective description, operator representation, locally convex properties by operators, (QNo)’, (LB)- and (LF)-cases
1Math. Subject Classification 2000 : primary 46E10, secondary 30H05, 32A70, 46A04/08/11/13/20/30/32/45, 46E40, 46M40
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About: Klaus. D. Bierstedt
Universität Paderborn, FB 17, Math., D-33095 Paderborn, Germany, klausd@uni-paderborn.de , http://math-www.uni-paderborn.de/~klausd/index.html