Natural generalized invertibility and prescribed idempotents

We study the natural inverse introduced by X. Mary and show some connections with the $(p,q)$-inverses of Djordjevic and Wei, where $p$ and $q$ are prescribed idempotents. We deal first with rings with identity and then specialize to the particular case of the algebra of bounded linear operators. We give a characterization of the set of operators along which an operator is natural invertible in terms of prescribed range and nullspace. Finally, the special case when the prescribed idempotent is the spectral projection is discussed.