Composition operators on spaces of real analytic functions
by
Pawel Domanski
A. Mickiewicz University (Poznan) and Institute of Mathematics, Polish Academy of Sciences
Coauthors: Michael Langenbruch (Universität Oldenburg)

We consider the classical space (algebra) A(\Omega) of real analytic functions on an open set \Omega subset or equal R^d equipped with its natural topology. We observe that algebra homomorphisms T:A(\Omega₁) --> A(\Omega₂) are automatically continuous and they are exactly composition operators C_j, C_j(f)=f o j, j:\Omega₂ --> \Omega₁ analytic map.

We characterize when C_j is a topologically isomorphic embedding. We show that for any two connected open sets \Omega₁, \Omega₂ subset or equal R^d there is a topological embedding C_j:A(\Omega₁) --> A(\Omega₂). We also embed into A(R) (via composition operators) both the space H(D) of holomorphic functions on the open unit disc and its dual H(K) - the space of germs of holomorphic functions on the closed unit disc K. The latter result allows to characterize for arbitrary open \Omega all Fréchet and all LB-spaces embedded into A(\Omega) as well as to characterize when one can embed isomorphically (as a locally convex space or as a topological algebra) one space A(\Omega) into another space of the same type.

(T)

Date received: April 10, 2000