# Morphisms of Banach spaces

What is the standard name in English for bounded linear maps $f:E\to F$ between Banach spaces such that the kernel $\ker(f)$ has a complement, and $\text{im}(f)$ is closed, and has a complement?

Apparently, when $f$ is injective or surjective, it is sometimes referred to as an admissible monomorphism or epimorphism, respectively. Is this terminology standard?

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In Helemskii's work on Banach (Co)Homology, this is the terminology he uses. I would consider it standard, in the sense that I'd use it, but I think I'd always be tempted to define what was meant. Theo's answer suggests that it's not standard if your audience is sufficiently large. –  Matthew Daws Feb 24 '11 at 9:08
Dear Ezra, As a somewhat related terminological remark, I think that Bourbaki (in his book Topological vector spaces) uses the terminology strict for 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 it is standard for number theorists working with $p$-adic Banach spaces. Regards, Matt –  Emerton Feb 24 '11 at 12:53
Seconding Emerton's comment: in categories with kernels and cokernels (and hence images and coimages) I have seen strict morphisms defined as those $f$ for which the canonical map from Coimage(f) to Image(f) is an isomorphism. (I came across this in work of Schneiders and Prosmans, but I don't know where the term originated.) –  Yemon Choi Feb 24 '11 at 20:07
@Emerton, @Yemon: Since this topic was brought up, I think if anyone is to be credited for this notion then it should be A. Weil who, as far as I know, introduced them in his book on integration in topological groups in the mid-thirties. –  Theo Buehler Feb 24 '11 at 22:38

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).

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the condition that there exist a $g$ with $fgf=f$ and $gfg=g$ is sometimes called regularity, by analogy with von Neumann regular rings. I have also heard such an $f$ called split since, if $f$ is epi then this condition is equivalent to $f$ being split epi, while if $f$ is mono it is equivalent to $f$ being split mono. –  Steve Lack Feb 24 '11 at 9:05
I've seen the term split used for those $f$ for which there exists $g$ satisfying $fgf=f$ (this is used in Weibel's Homological Algebra book for instance) –  Yemon Choi Feb 24 '11 at 20:03