Geodesic
Spectral mapping theorem for the Davies–Helffer–Sjöstrand functional calculus
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 8 (2012) no. 3, pp. 221-239 Cet article a éte moissonné depuis la source Math-Net.Ru

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We give a direct non-abstract proof of the spectral mapping theorem for the Davies–Helffer–Sjöstrand functional calculus for linear operators on Banach spaces with real spectra and consequently give a new non-abstract direct proof for the spectral mapping theorem for self-adjoint operators on Hilbert spaces. Our exposition is closer in spirit to the proof by explicit construction of the existence of the Functional Calculus given by Davies. We apply an extension theorem of Seeley to derive a functional calculus for semi-bounded operators. 
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N. S. Claire. Spectral mapping theorem for the Davies–Helffer–Sjöstrand functional calculus. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 8 (2012) no. 3, pp. 221-239. http://geodesic.mathdoc.fr/item/JMAG_2012_8_3_a0/

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