Past exam of the stochastic analysis and partial differential equations course of the University of Oxford / 2025 / 3 / a / ii

For $\lambda \ne \mu$, apply the resolvent identity from part (i) in both orders:The two quotients are equal. For $\lambda =\mu$ the assertion is immediate. HenceThus resolvent operators from the same strongly continuous semigroup commute. This conclusion does not require commuting arbitrary bounded linear operators; it uses the fact that both arise from the same semigroup generator.