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Wikipedia calls Resolvent formalism a useful tool for relating complex analysis to studying the spectra of a linear operator on a Banach space. Sure, I believe you because I've seen results that use the resolvent in the proof.

I also recognize that the resolvent of T encodes T's eigenvalues as poles.

Granting all that, when should I look at a problem and think "oh, of course, I should translate this problem into a resolvent formulation", and more generally, is it just because it has come in handy and the aforementioned fact that the resolvent is useful? Or is there some intuition that I'm missing?

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