In this paper we extend to an infinite dimensional setting some results on the Shadowing property that are known on finite dimensional compact manifolds without border and in \(\mathbb {R}^n\) . In fact, we show that if \(\{{\mathcal {T}}(t):t\ge 0\}\) is a Morse-Smale semigroup defined in a Hilbert space, with global attractor \(\mathcal {A}\) and non-wandering set given by its equilibria, then \({\mathcal {T}}(1)|_{\mathcal {A}}:\mathcal {A}\rightarrow \mathcal {A} \) admits the Lipschitz Shadowing property. Moreover, for any positively invariant bounded neighborhood \(\mathcal {U}\supset \mathcal {A}\) of the global attractor, the map \({\mathcal {T}}(1)|_{\mathcal {U}}:\mathcal {U}\rightarrow \mathcal {U}\) has the Hölder–Shadowing property. As applications, we obtain results related to the structural stability of Morse-Smale semigroups, that were only known on finite dimension, and continuity of global attractors.
