TAILIEUCHUNG - Báo cáo toán học: "Functional calculus with sections of an analytic space "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Chức năng tính toán với các bộ phận của một không gian phân tích. | Copyright by INCREST 1980 J. OPERATOR THEORY 4 1980 297-306 FUNCTIONAL CALCULUS WITH SECTIONS OF AN ANALYTIC SPACE MIHAI PUTINAR The aim of this paper is to give an algebraic-topological approach to the representation theory of algebras of global sections of an analytic space in connection with the functional calculus problem for finite commuting systems of linear continuous operators on a Fréchet space. The paper contains as a particular case the existence of the analytic functional calculus for commuting systems of regular operators on a Frechet space as well as a spectral mapping theorem. The analytic space context yields a more refined functional calculus even for finite systems of commuting operators namely a functional calculus which takes not only the spectrum but also analytic relations satisfied by the operators into account. Moreover the more general case of representations of algebras of global sections of a Stein space does not reduce in general to the case of finite systems of operators. Finally working with sheaves of -functions we obtain an analogue of the theory which corresponds to generalized spectral systems of commuting operators. Our technique is based on a relative homology theory in the sense of 5 Cap. IX for topological algebras. Such a theory was developed for the proof of Grauert s coherence theorem see for example 1 11 and afterwards independently a homology theory essentially the same was elaborated and used by J. L. Taylor in the functional calculus theory. Let us recall some notations and terminology. Let A be a nuclear Frechet C-algebra. In the relative homology theory which we shall use the free -modules are of the form A E where E is a Frechet space and the admissible short exact sequences of Frechet yl-modules are that which are topologically C-split. Each Frechet yt-module M has a natural free resolution the Bar resolution 0 - M-Í- A M - A A M . . The tensor product of two Frechet J-moduIes M and N is the quotient space of M N by

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