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've also read bits of Kato's Perturbation Theory for Linear Operators where he says it simplifies proofs.

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 of the aforementioned fact that the resolvent is useful? Or is there some intuition that I'm missing?

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've also read bits of Kato's Perturbation theory for linear operators where he says it simplifies proofs.

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 of the aforementioned fact that the resolvent is useful? Or is there some intuition that I'm missing?