Holomorphy in locally convex spaces and operators on the Fock spaces

Abstract

Let E be a complex strong dual of a Fréchet-Schwartz space and let F be a Fréchet space ; let H be a closed hyperplan of E ; for each n ε ℕ let u_n be a n-multilinear symmetric continuous map from Eⁿ to F (and u_o ε F); it is proved in § 3 that there exists a F-valued holomorphic map defined in the complementary of H such that f admits the (possibly divergent) formal series as an asymptotic expansion at the origin of E (in a sense made precise below); this result is different but somewhat similar to the theorem of [4]. Applications of these results to formal power series of operators on the Fock spaces of Quantum Fields are sketched at the end of the paper.