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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các tài sản chồng chất cho phép tính chức năng của Taylor. | J. OPERATOR THEORY 7 1982 149-155 Copyright by INCREST 1982. THE SUPERPOSITION PROPERTY FOR TAYLOR S FUNCTIONAL CALCULUS MIHAI PUTJNAR This note is a continuation of 10 and it contains some functorial properties of the functional calculus with sections of an analytic space developed there. The spectral mapping theorem 12 and the superposition theorem in several variables spectral theory 1 4 are particular cases of Theorem 1 respectively Theorem 2. The implicit function theorem of Arens and Calderon f l 6 is extended to Taylor s joint spectrum. The proofs are based on the topological-homological techniques initiated by J. L. Taylor in 14 and presented shortly in the form used in this paper in 10 The reader is refered to 2 8 and 13 for a more complete description of this theory. Some facts from several complex variables function theory like Cech cohomology a Kiinneth formula and the existence of the envelope of holomorphy for an open subset of C are also used in this paper. No integral representation formulas for analytic functions appear explicitly in the proofs ft is interesting to remark that the superposition theorem for Taylor s functional calculus Theorem 2 below need for the proof a coordinateless argument by passing from an open subset of C to its envelope of holomorphy which is a Stein manifold. This may be the substitute for Arens and Calderon s Lemma from the classical spectral theory. Parts of this paper have appeared in ÍNCREST Preprint Series in Mathematics No. 43 1979 No. 39 1980 and No. 3 I98J. First of all let US recall some notation and results from 10 Let A be an analytic finite dimensional Stein space and let M be a Frechet ỡ X -module. Then there exists a closed subset ơ x M of X with the following properties a A Stein open subset V of X is disjoint of ơ X M if and only if T X 0 F M 0 for each q 0 10 Proposition 3.a. . 150 MIHAI PL IIN AR b If z is a Stein covering of ơ X M with finite dimensional nerve then Tor w ÝT M s M for q 0 and 0 for q 0 .
