Spectral theory and sheaf theory. II
The paper treats with sheaf theoretic methods the local spectral theory, the duality between spectral decompositions and the Fredholm theory of commutative n-tuples of linear operators acting on Fréchet spaces. Originating in the landmark paper ''A general framework for a multi- operator functional calculus'', Adv. Math. 9, 183-252 (1972; Zbl 0271.46041) by \textit{J. L. Taylor}, our approach starts with the observation that the n-tuples which are appropriate for local spectral theory may be modelled as the multiplications with the coordinate functions on the global sections space of a Fréchet analytic sheaf on \({\mathbb{C}}^ n\), called the sheaf model. More specifically, the sheaf model of an n-tuple a of operators on a Fréchet space X, which satisfies a cohomological condition named (\(\beta)\) is: \[ {\mathcal F}(U)={\mathcal O}(U,X)/\sum^{n}_{i=1}(z_ i-a_ i){\mathcal O}(U,X),\quad U=\overset\circ U\subset {\mathbb{C}}^ n. \] Here \({\mathcal O}(U,X)\) stands for the Fréchet space of all analytic X-valued functions defined on U. This relationship turns out to be benefit for both the multi-dimensional spectral theory as well as for the theory of analytic non-coherent sheaves. Thus the sheaf model carry in a natural way the spectral information concerning important classes of n-tuples. For instance some new examples of subnormal n-tuples of operators are presented by these methods in the last part of the paper. On the other hand, it is proved in the same time a general sheaf theory result which asserts that any analytic Fréchet soft sheaf on a Stein space is quasi-coherent (in the terminology of the French school). The paper is the second in a series of three articles with the same title, as follows: I. Oper. Theory Adv. Appl. 11, 283-297 (1983; Zbl 0525.47010) and III. (in cooperation with \textit{J. Eschmeier}) J. Reine Angew. Math. 354, 150-163 (1984; Zbl 0539.47002).
duality between spectral decompositions and the Fredholm theory of commutative n-tuples of linear operators acting on Fréchet spaces
scientific article; zbMATH DE number 3983909
scientific article