Let $T : X \rightarrow Y$ be a bounded linear operator between spaces $X$ and $Y$. Suppose that the range $T(X) \subseteq Y$ is closed.
We know that bounded linear operators have closed kernel, and so $N := \ker T$ is closed. But is $N$ a complemented closed subspace? That means that there exists $M \subseteq X$ closed such that $X \simeq M \oplus N$ as Banach spaces?
The converse implication is false, since otherwise any injective operator would have closed range, but there are counterexamples for that.
Otherwise, can you give a counterexample?
