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Spectral decompositions and analytic sheaves. (English) Zbl 0855.47013

London Mathematical Society Monographs. New Series. 10. Oxford: Oxford Univ. Press. x, 362 p. (1996).
The book presents an up to date picture of the “analytic” ideology (going back to work by E. Bishop and J. L. Taylor) in the problems, related to the functional calculus and decomposability of \(n\)-tuples of commuting operators on Fréchet and Banach spaces.
The first chapter has an introductory character. In Chapter 2-5 the functional calculus of \(n\)-tuples of commuting operators on Fréchet spaces is worked out. First in Chapter 2 the functional calculus is developed by means of Cauchy-Weil integral representations. Then in Chapter 3-5 there is described the necessary topological homology theory and the functional calculus is constructed by means of the obtained apparatus. In Chapter 6 the analytical ideology is applied to the problems related to the decomposition of \(n\)-tuples of commuting operators on Banach spaces. The results here present the Bishop’s type theory for \(n\)-tuples of operators.
The last four chapters (7-10) contain various applications of the theory. Among them are results related to Corona theorem, spectral analysis of \(n\)-tuples of analytic Toeplitz operators acting on Bergman spaces over bounded pseudoconvex domains in \(\mathbb{C}^n\), multidimensional index theory and others.
Reviewer: V.Moroz (Minsk)

MSC:

47A60 Functional calculus for linear operators
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32C35 Analytic sheaves and cohomology groups
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