By definition, a bounded linear operator $T: X \to X$ has a non-trivial closed invariant subspace W on a Banach space $X$ if for every vector $\omega \in W$, $T(\omega)$ belongs to $W$ (i.e, $T(W) \subset W$).
Could anyone help me with an example of applications of an operator that has a nontrivial closed invariant subspace?