 Define "e"("z") = 1 if "z" ∈ "U" and "e"("z") = 0 if "z" ∈ "V".Then "e" is a holomorphic function with ["e"("z")]2 = "e"("z") and so, for a suitable contour Γ which lies in "U" ∪ "V" and which encloses σ("T"), the linear operator

will be a bounded projection that commutes with "T" and provides a great deal of useful information.It transpires that this scenario is possible if and only if "K" is both open and closed in the subspace topology on "σ"("T").