On the connected components of the conjugacy class of projectors on ℓp⊕ℓq

We characterize the projectors P on a Banach space E having the property of being connected to all the other projectors obtained as a conjugation of P. Using this characterization we show an example of a Banach space where the conjugacy class of a projector splits into several path-connected components, and describe the conjugacy classes of projectors onto subspaces of ℓp⊕ℓq with p≠q.