I know that if $V$ is a finite-dimensional vector space and $T$ is a linear operator on $V$, then $T$ is a direct sum of a nilpotent operator and an invertible operator.
My question is: Is there an infinite-dimensional vector space $V$ over a field $\mathbb{F}$ such that every linear operator $T$ on $V$ can be written as a direct sum of a nilpotent linear operator and an invertible linear operator?
I mean "There are subspaces such that $V=M\oplus N$ where $T$ is invertible on $M$ and $T$ is nilpotent on $N$"?