I don't know if there's a universally accepted name for this notion. Of course, if the kernel and the complement of the image are finite-dimensional, such a map is a Fredholm operator. I would suggest to call them pseudo-invertible (in reference to the Moore-Penrose pseudo-inverse), because $f:E \to F$ has complemented kernel and image if and only if there is a $g:F \to E$ such that $fgf = f$ (and $gfg = g$).

Since the question is tagged homological algebra, let me point out that these morphisms are precisely the morphisms factoring as $E \twoheadrightarrow I \rightarrowtail F$, where $\twoheadrightarrow$ stands for a split epimorphism and $\rightarrowtail$ for a split monomorphism. These are the *admissible monomorphisms* and *admissible epimorphisms* for the *split exact structure* (in the sense of Quillen) on the additive category of Banach spaces and bounded linear maps, see here for more on this. However, there are several exact structures on the category of Banach spaces (at least three of interest), so admissible monomorphism/epimorphism alone is not good enough from this point of view.

I've seen several names for the maps you're asking about in the literature:

In Borel-Wallach, *Continuous cohomology, discrete subgroups and representations of reductive groups*, Annals of Math. Studies **94**, Princeton University Press (1980), IX 1.5, they are called $s$-*morphisms* (actually for Hausdorff locally convex spaces).

Guichardet, *Cohomologie des groupes topologiques et des algèbres de Lie*, Textes Mathématiques **2**, Fernand Nathan, Paris (1980), Appendice D, Def. D.1 calls them *strong* (again in the setting of Hausdorff locally convex spaces).

Monod, *Continuous Bounded Cohomology of Locally Compact Groups*, Springer Lecture Notes in Mathematics, **1758** (2001), Definition 4.2.2, calls them *weakly admissible* (the weakly refers to the fact that his notion of admissibility requires that if $f$ is of norm $1$, there is a pseudo-inverse $g$ of norm $1$, at least for monomorphisms).

Topological vector spaces) uses the terminologystrictfor a morphism with closed image. I don't know if this continues to be standard terminology for current researchers on archimedean Banach spaces, but I can say that itisstandard for number theorists working with $p$-adic Banach spaces. Regards, Matt $\endgroup$