resolvent

When lambda is NOT in the spectrum, the equation (T - lambda*I)x = y can be solved uniquely for x in terms of y, and the solving operation is itself a bounded operator. That solving operator is the resolvent. It is the explicit inverse that exists exactly where the operator is well-behaved, and its analytic dependence on lambda is the engine behind almost all of spectral theory.

For lambda in the resolvent set, define R(lambda) = (lambda*I - T)^{-1} (sign conventions vary). The resolvent is a bounded operator, and as a function of lambda it is analytic (holomorphic, in the operator-valued sense) on the resolvent set. It satisfies the resolvent identity R(lambda) - R(mu) = (mu - lambda) R(lambda) R(mu), which encodes how the inverse varies and shows that R commutes with itself at different parameters.

Because R(lambda) is analytic off the spectrum and must blow up as lambda approaches the spectrum, complex-analytic tools — Cauchy's theorem, Liouville's theorem, power series — transfer directly to operators. This is exactly how one proves the spectrum is nonempty and builds the holomorphic functional calculus by integrating functions against the resolvent.