  Given an idempotent $p$ in a Banach algebra and following the study in
\cite{P50} of p-invertibility, we consider here left p-invertibility, right
p-invertibility and p-invertibility in the Calkin Algebra $\mathcal{C}(X),$
where $X$ is a Banach space. Then we define and study left and right
generalized Drazin invertibility and we characterize left and right Drazin
invertible elements in the Calkin algebra. Globally, this leads to define and
characterize the classes of P-Fredholm, pseudo B-Fredholm and weak B-Fredholm
operators.
