L-shadowing lemma for the Cauchy equation

We prove that if the Cauchy problem $\dot{u}=Au$ in a Banach space is hyperbolic, then the problem has the L-shadowing property. Conversely, if the space is finite-dimensional and the L-shadowing property is satisfied, then the problem is hyperbolic. This generalizes a previous result by Ombach \cite{o, o1} for linear homeomorphisms. Some short applications are given.