Does $\ker T\cap {\rm Im}\,T=\{0\}$ imply $V=\ker T\oplus{\rm Im}\,T$

Question

Let $T: V\rightarrow V$ be a linear operator of the vector space $V$.

We write $V=U\oplus W$, for subspaces $U,W$ of $V$, if $U\cap W=\{0\}$ and $V=U+W$.

If we assume $\dim V<\infty$, then by the rank-nullity theorem, $\ker T\cap {\rm Im}\,T=\{0\}$ implies $V=\ker T\oplus {\rm Im}\,T$.

However, my question is about the case $\dim V$ is infinite. Is it still true? What if $T$ has a minimal polynomial?

Thanks.

Answer 1

Consider the shift operator $s$, defined on $\text{Vect}(e_i, i\in\mathbb{N})$, where $s(e_n)=e_{n+1}$ for $n\in\mathbb{N}$. Note that $\ker(s)=0$ but $s$ is not surjective.