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Jul 6, 2020 at 13:15 comment added Wahome The route we took to functional calculus was via Helffer-Sjostrand, not the 'traditional' spectral theorem. But I see now what to do, thank you!
Jul 6, 2020 at 12:21 comment added Nik Weaver "Why are we allowed to identify operators constructed by these two techniques?" The short answer is: the spectral theorem.
Jul 6, 2020 at 11:54 comment added Paul Siegel For me these two methods are the same: the spectral theorem asserts that self adjoint operators are unitarily equivalent to multiplication operators, and this allows us to define an action of a suitable algebra of functions i.e. a functional calculus. Did you have something else in mind?
Jul 6, 2020 at 10:27 history asked Wahome CC BY-SA 4.0