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Fréchet algebra sheaf cohomology and spectral theory
by
Anar Dosiev
Institute of Mathematics and Mechanics, Azerbaijan
The talk is devoted to a multi-operational functional calculus problem with respect to a noncommutative Fréchet algebra sheaf. Our approach generalizes Taylor-Helemskii-Putinar framework of holomorphic functional calculus for a commutative operator family. Using the sheaf cohomology technique and methods of topological homology, we propose a functional calculus on a neighborhood of Putinar spectrum \sigma(S, X) of a Fréchet module X over the algebra A(=S(\Omega)) of all global sections with respect to a Fréchet algebra sheaf S. The ground space \Omega of the considered sheaf S satisfies usual demands like to be acyclic with respect to the sheaf S.
As an application of this calculus it is considered a sheaf Tg of germs of formally-radical functions in elements of of a finite-dimensional nilpotent Lie algebra g. In this case the known Taylor spectrum \sigma(g, X) of a Fréchet g-module X belongs to Putinar spectrum \sigma(Tg, X) and they coincide for a Banach g-module X. In particular, we prove a holomorphic functional theorem on a neighborhood of Taylor spectrum of a Banach space operators generating a supernilpotent Lie subalgebra. The latter result is a noncommutative generalization of Taylor functional calculus.
Date received: May 20, 2004
Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cane-33.