the resolvent operator

The resolvent is the Laplace transform of the semigroup, and it is the workhorse that converts the dynamic, differential object A (the generator) into a friendly bounded operator one can actually compute with. Where the generator is unbounded and lives only on a domain, the resolvent R(lambda) is bounded and defined everywhere, so much of semigroup theory is conducted through it. Probabilistically it is the operator of 'exponentially discounted total reward'.

Definition: for lambda > 0 (more generally in the resolvent set), R(lambda) = (lambda I - A)^(-1). Equivalently, and this is the key identity, it is the Laplace transform of the semigroup, R(lambda) f = integral_0^infinity e^(-lambda t) P_t f dt. Reading the right-hand side probabilistically, (R(lambda) f)(x) = E_x[ integral_0^infinity e^(-lambda t) f(X_t) dt ] is the expected total of the reward f collected along the path, discounted at rate lambda. From the contraction property ||P_t|| <= 1 one gets ||R(lambda)|| <= 1/lambda. The resolvent is injective with range D(A), satisfies A R(lambda) = lambda R(lambda) - I and R(lambda) A f = lambda R(lambda) f - f on D(A), and recovers the generator in the limit lambda R(lambda) f -> f as lambda -> infinity; in fact A f = lim lambda (lambda R(lambda) f - f). Different lambda are tied together by the resolvent equation.

Why it matters: the resolvent is how Hille-Yosida is stated and proved (its bound is the generation criterion, and the semigroup is rebuilt from it), how one defines functions of the generator and analyses spectrum, and how stationary problems are solved: (lambda I - A) u = f is the resolved/discounted version of the evolution equation. The discounting is essential — without the factor e^(-lambda t) the time integral of P_t f generally diverges, so lambda > 0 is not optional; it is exactly the discount rate that makes the lifetime reward finite and the inverse bounded.