How do I show a left inverse of a bounded linear operator on Banach space?

Question

If $A$ is a bounded linear operator on a Banach space X, with a left inverse $A_l^{-1}$, and P is a projection (also on X), how do I show that $A_l^{-1}P$ is also a left inverse of A (i.e. $A_l^{-1}PA=I$, the identity operator)?

Answer 1

In general and without further assumptions, $A_l^{-1}PA\neq I$. Take $A\neq0$ for which some $A_l^{-1}$ exists. And take the projection of $X$ onto the subspace $\{0\}$. Then $A_l^{-1}P=0$. (So you need to assume $P$ is a projection onto a subspace that incorporates $A(X)$).