I hope this is not too naive a question for MO. I've been taking a mathematical physics course, and was shown how operators like $\sqrt{1-\Delta}$ could be defined by taking multiplication operators in Fourier space. Later it emerged that functional calculus methods also allow one to construct similarly general operators, and it seems that they are considered equivalent, as some texts do not even specify how they interpreted such formulas when they write them. Why are we allowed to identify operators constructed by these two techniques (if we are)?
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3$\begingroup$ 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? $\endgroup$– Paul SiegelCommented Jul 6, 2020 at 11:54
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3$\begingroup$ "Why are we allowed to identify operators constructed by these two techniques?" The short answer is: the spectral theorem. $\endgroup$– Nik WeaverCommented Jul 6, 2020 at 12:21
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1$\begingroup$ 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! $\endgroup$– WahomeCommented Jul 6, 2020 at 13:15
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