Representation of Generalized Bi-Circular Projections on Banach Spaces

We prove several results concerning the representation of projections on arbitrary Banach spaces. We also give illustrative examples including an example of a generalized bi-circular projection which can not be written as the average of the identity with an isometric reflection. We also characterize generalized bi-circular projections on $C_0(\Om,X)$, with $\Om$ a locally compact Hausdorff space (not necessarily connected) and $X$ a Banach space with trivial centralizer.