Linear operator with non-overlapping kernel and range spaces

Question

Let $V$ be a vector space and $T\colon V\rightarrow V$ a linear operator with $\ker T\cap T(V)=\{0\}$. Then what can one say about $T^2$? If $V$ is finite-dimensional, then one can show that the rank of $T$ and that of $T^2$ are equal. But in the case of infinite- dimensional $V$, is $T$ a specific linear operator? what propery does T necessarily have? Thanks for your help.