Consider the category Ban of Banach spaces with linear contractions. I want to find a characterization of the split monomorphisms and of the split epimorphisms in this category, however I couldn't make much progress.
I know that every split monomorphism is a monomorphism in any category and in Ban these are precisely the injective morphisms, but it seems that not any injective linear contraction is also a split monomorphism. In fact, I am kind of confused. Let $f:X\to Y$ be an injective linear contraction. Then, we can show that it has a linear left inverse, call it $g:Y\to X$. Now we have $1_X=g\circ f$, which implies that $1=||g\circ f||\le ||g||\cdot ||f||\le ||g||$. I guess that this means that $g$ must have norm $1$ in order for it to be a contraction. But I don't know what other condition I should impose on $f$ so that such a $g$ exists.
Similarly, every split epimorphism is an epimorphism, so this means that every split epimorphism in Ban has a dense image, but again I don't think the converse holds and I don't know how to find what other properties I should add.
As requested, this is my comments turned into an answer, with somewhat more details added:
$f$ is a split monomorphism iff it is an isometric embedding and that its range is complemented in $Y$ and admits a norm 1 projection (from $Y$ to the range of $f$). Indeed, if $f$ is a split monomorphism, then there exists a linear contraction $g: Y \rightarrow X$ such that $g \circ f = Id_X$. Then $||x|| \geq ||f(x)|| \geq ||g(f(x))|| = ||x||$, so $||x|| = ||f(x)||$ and $f$ is an isometric embedding. (This argument also shows that in Ban, isomorphisms are isometric linear bijections.) $f \circ g$ then is a norm 1 projection from $Y$ onto the range of $f$ and the range is complemented by $\mathrm{ker}(g)$ in $Y$. Conversely, if $f$ is a linear embedding and there is a norm 1 projection $P$ from $Y$ onto the range of $f$, then we may let $g = f^{-1} \circ P$, which one can show is a linear contraction and that $g \circ f = Id_X$, so $f$ is a split monomorphism.
By similar arguments, split epimorphisms are exactly norm 1 projections onto closed subspaces followed by isometric isomorphisms. Furthermore, if you allow your category to contain all bounded linear maps, similar arguments show that split monomorphisms are embeddings onto closed, complemented subspaces; and split epimorphisms are projections onto closed subspaces followed by bounded isomorphisms. (These require the use of the open mapping theorem.)