When is the sum of complemented subspaces complemented?

TL;DR

This paper establishes a sufficient condition for the sum of complemented subspaces in a Banach space to be complemented, provides a projection formula, and explores applications including stability and sum of tensor powers.

Contribution

It introduces a sharp sufficient condition for the sum of complemented subspaces to be complemented and derives applications in Banach space theory and tensor products.

Findings

A new sufficient condition for complementability of sums of subspaces.

A formula for the projection onto the sum of subspaces.

Results on stability of complementability under perturbations.

Abstract

We provide a sufficient condition for the sum of a finite number of complemented subspaces of a Banach space to be complemented. Under this condition a formula for a projection onto the sum is given. We also show that the condition is sharp (in a certain sense). As applications, we get (1) sufficient conditions for the complementability of sums of marginal subspaces in $L^p$ and sums of tensor powers of subspaces in a tensor power of a Banach space and (2) quantitative results on stability of the complementability property of the sum of linearly independent subspaces.