Segments of bounded linear idempotents on a Hilbert space

Let H be a separable Hilbert space. We prove that any two homotopic idempotents in the algebra L(H) may be connected by a piecewise affine idempotent-valued path consisting of 4 segments at most. Moreover, we show that this constant is optimal provided H has infinite dimension. We also explain how this result is linked to the problem of finding common complements for two closed subspaces of H.