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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 ...

I'm not an analyst, so forgive me if what I'm asking is not suitable for Mathoverflow. For convenience, let $X$ be a compact complex manifold, and $E$ a holomorphic vector bundle on $X$. Let $H$ be ...