Path connectivity of k-generalized projectors

Let B(H) be the set of all bounded linear operators on a Hilbert space H. An operator A∈B(H) is said to be a k-generalized projector if Ak=A∗, where k⩾2 is an integer and A∗ denotes the adjoint of A. Denote by B(H)k-GP the set of all k-generalized projectors in B(H). In this paper, we show that any two homotopic k-generalized projectors are path connected and that there does not exist a segment [P,Q]⊆B(H)k-GP when P and Q are two different k-generalized projectors.