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Can someone give me an example of a Banach space $X$ and contractive projection $P\in\mathcal{B}(X)$ such that $\ker P$ is not a range of any contractive projection $Q\in\mathcal{B}(X)$?

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The projection from the space $c$ of convergent sequences to the $1$-dimensional subspace of constant sequences with kernel $c_0$. Any projection $c \to c_0$ has norm at least $2$ (cf. exercises to chapter 2.5 in Albiac-Kalton).

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