 Laplace-transform formula for a semigroup resolvent

ID: laplace-transform-formula-for-a-semigroup-resolvent

A

generates a C0-semigroup satisfying

∥ U (t) ∥ \leq M e^{ω t}

, then for

Re z > ω

(z I - A)^{- 1} x = \int_{0}^{\infty} e^{- z t} U (t) x d t .

Repeated differentiation gives

n! (z I - A)^{- (n + 1)} x = \int_{0}^{\infty} t^{n} e^{- z t} U (t) x d t

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