Atlas: Operator representations for spaces of vector valued holomorphic functions by Klaus D. Bierstedt

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1999 Summer Conference on Topology and its Applications
August 4-7, 1999
C.W. Post Campus of Long Island University
Brookville, NY, USA

Organizers
Sheldon Rothman, Ralph Kopperman

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Operator representations for spaces of vector valued holomorphic functions
by
Klaus D. Bierstedt
Univ. Paderborn (Germany)
Coauthors: Silke Holtmanns

For weighted spaces HV(G) of holomorphic functions with a topology stronger than uniform con-
vergence on the compact subsets of G (open in C^N) and for quasibarrelled E, the following spaces
are topologically isomorphic:

HV(G, E'_b) = L_b(E, HV(G)) and HV₁(G₁, HV₂(G₂)) = H(V₁ \otimesV₂)(G₁ ×G₂),

cf. [1]. Starting from these formulas, we discuss the problem when similar isomorphisms hold for weighted inductive limits VH(G) = ind_n Hv_n(G) of spaces of holomorphic functions. It turns out
that this is a completely different question, related to a special case of Grothendieck's ``Problème
des topologies'' and to the vector valued projective description problem. A positive result is derived, and some examples are given.

[1] Klaus D. Bierstedt, Silke Holtmanns, An operator representation for weighted spaces of vector valued holomorphic functions, to appear in Results Math.

Date received: June 5, 1999