the resolvent equation
The resolvent equation (or first resolvent identity) is the algebraic relation that ties together the resolvents at different values of the spectral parameter lambda. It looks like a small computational fact, but it is what makes the resolvent a coherent, analytically well-behaved family — in particular it shows the resolvent is a smooth (indeed analytic) operator-valued function of lambda, which is the engine behind functional calculus and spectral theory.
Statement: for lambda, mu in the resolvent set, R(lambda) - R(mu) = (mu - lambda) R(lambda) R(mu). The derivation is pure algebra: write R(lambda) - R(mu) = R(lambda) [ (mu I - A) - (lambda I - A) ] R(mu) = R(lambda)(mu - lambda) R(mu). Two consequences follow immediately. First, resolvents at different lambda COMMUTE: R(lambda) R(mu) = R(mu) R(lambda), because the right-hand side is antisymmetric in lambda, mu in a way that forces it. Second, dividing by (mu - lambda) and letting mu -> lambda shows R is differentiable in lambda with d/dlambda R(lambda) = - R(lambda)^2, and by iteration the n-th derivative is (-1)^n n! R(lambda)^(n+1); hence lambda -> R(lambda) is analytic on the resolvent set with a convergent power-series (Neumann-type) expansion R(mu) = sum (lambda - mu)^n R(lambda)^(n+1).
Why it matters: analyticity of the resolvent is the foundation of holomorphic functional calculus (defining f(A) by a contour integral of f(z) R(z)), of perturbation theory for operators (how eigenvalues move when A is perturbed), and of inverting the Laplace transform to recover P_t from R(lambda). In the Markov setting the identity expresses a probabilistic restart: discounting at rate lambda then re-discounting at rate mu is consistent with the path being run once, which is why the two-parameter family collapses to a one-parameter flow. The identity is purely algebraic and requires only that both lambda and mu lie in the resolvent set — no extra regularity of A beyond that is needed.
Differentiating R(lambda) for Brownian motion confirms the identity: with R(lambda) the convolution by (1/sqrt(2 lambda)) e^(-sqrt(2 lambda)|x|), one checks directly that R(lambda) - R(mu) = (mu - lambda) R(lambda) R(mu), and d/dlambda R(lambda) = -R(lambda)^2, matching the abstract result without any reference to the explicit kernel.
The resolvent identity makes lambda -> R(lambda) analytic, the gateway to functional calculus and perturbation theory.
The identity needs only lambda, mu in the resolvent set and is what forces resolvents at different lambda to commute and R(lambda) to be analytic in lambda.