It is known (Lindenstrauss, Tzafriri, On the complemented subspaces problem) that a real Banach space all of whose closed subspaces are complemented (i.e. have a closed supplement) is isomorphic (as a tvs) to a Hilbert space. But I am interested in complementing a special kind of subspaces: subspaces F of a Banach space E satisfying either of the following two equivalent conditions:
(i) $E/F \simeq F$
(ii) $\exists f \in \mathcal{L}(E) \quad F = \ker f = im f$
Are such "half-dimensional" subspaces always complemented? Is there a class of spaces (larger than that of Hilbert spaces) where this is true?
Actually, (i)=>(ii) is true for any tvs, and (ii)=>(i) is true for complete metrizable spaces (over nondiscrete valued fields). Under that same condition, two closed algebraic supplements are topological supplements (from the open mapping theorem). So the question is still interesting in that setting. By the way, is the Lindenstrauss–Tzafriri theorem true for such spaces (or any class larger than that of real Banach spaces — or real Fréchet spaces, as the mathscinet review says)?
Best Answer
I believe the answer to your first question is no. The counterexample I have in mind is related to the peculiar fact (first proved by Enflo, Lindenstrauss and Pisier) that being isomorphic to a Hilbert space isn't a "three-space property". More specifically, there is an example of a Banach space $E$ with an uncomplemented subspace $F$ such that both $F$ and $E/F$ are isomorphic to $\ell^2$. You can also find examples where $F$ and $E/F$ are isomorphic to any $\ell^p$. This is due to Kalton and Peck.