On the connected components of the conjugacy class of projectors on ell_poplusell_q

We characterize the projectors $ P $ on a Banach space $ E $ having the property of being connected to all the others projectors obtained as a conjugation of $ P $. Using this characterization we show an example of 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 $ \ell_p\oplus\ell_q $ with $ p\neq q $.