Parametrizing projections with selfadjoint operators

Let H=H+⊕H− be an orthogonal decomposition of a Hilbert space, with E+, E− the corresponding projections. Let A be a selfadjoint operator in H which is codiagonal with respect to this decomposition (i.e. A(H+)⊂H− and A(H−)⊂H+). We consider three maps which assign a selfadjoint projection to A:1.The graph map Γ: Γ(A)=projection onto the graph of A|H+.2.The exponential map of the Grassmann manifold P of H (the space of selfadjoint projections in H) at E+: exp⁡(A)=eiπ2AE+e−iπ2A.3.The map p, called here the Davis' map, based on a result by Chandler Davis, characterizing the selfadjoint contractions which are the difference of two projections.The ranges of these maps are studied and compared.Using Davis' map, one can solve the following operator matrix completion problem: given a contraction a:H−→H+, complete the matrix(⁎a/2a⁎/2⁎)to a projection P, in order that ‖P−E+‖ is minimal.