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The functional calculus. (English) Zbl 0858.47012
The formula of Helffer and Sjöstrand for functions \(f(H)\) of a selfadjoint operator \(H\) is generalized and put into the real perspective. The author constructs a functional calculus for Banach space operators with real spectra and slowly increasing resolvents when approaching the spectrum. This very well written paper with its strong didactic impetus will help that Helffer-Sjöstrand formula will become a standard tool in spectral theory. The power of this calculus is demonstrated by the same author in [J. London Math. Soc., II. Ser. 52, No. 1, 177-184 (1995)] concerning \(L^p\)-independence of spectra.
Reviewer: V.Wrobel (Kiel)

47A60 Functional calculus for linear operators
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