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Let $X$ be a Banach space.

Question. Suppose $X_1,...,X_n$ are complemented subspaces of $X$. When is the sum $X_1+...+X_n$ complemented? Further, suppose we know some projections $P_1,...,P_n$ onto $X_1,...,X_n$. Is there a formula for a projection onto $X_1+...+X_n$ (it terms of $P_1,...,P_n$) (of course, under certain conditions)?

Remark. I have some results on this question. However, the question seems to be a very basic question in the theory of complemented subspaces, so I have doubts about originality of my results. The only references I have found are results by Alan LaVergne (paper "Remark on sums of complemented subspaces"), Lars Svensson (paper "Sums of complemented subspaces in locally convex spaces"), Manuel Gonzalez (paper "On essentially incomparable Banach spaces", Lemma 1), and $S\ddot{u}leyman$ $\ddot{O}nal$, Murat Yurdakul (paper "On sums of complemented subspaces").

My results on the question are presented in my recent paper "When is the sum of complemented subspaces complemented?" https://arxiv.org/abs/1606.08048.

I would be very grateful for your comments.

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    $\begingroup$ You can find two closed subspaces of $\ell_2$ (hence each is complemented) and yet their sum is not closed. $\endgroup$
    – Bunyamin Sari
    Commented May 2, 2016 at 17:11
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    $\begingroup$ Other than the theorem of Edelstein-Wojtaszczyk (see LaVergne's paper) I do not know of anything that is not essentially trivial. $\endgroup$
    – Bill Johnson
    Commented May 2, 2016 at 20:19

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