Let $C(\mathbb{R})$ is the set of all continuous functions $\mathbb{R}\to\mathbb{R}$.
I want to find a linear operator $T:C(\mathbb{R})\rightarrow C(\mathbb{R})$, proper subspaces $W_{1}, W_{2}$ of $C(\mathbb{R})$ such that $C(\mathbb{R})=W_{1}\oplus W_{2}$ and $T|_{W_{1}}$ is nilpotent $( T|_{W_{1}}\neq 0)$ and $T|_{W_{2}}$ is invertible.
I've tried this one, but it didn't work.
Thanks for your kindly help.