  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 $.
