Self-adjoint block operator matrices with non-separated diagonal entries and their Schur complements

In this paper self-adjoint 2×2 block operator matrices A in a Hilbert space H1⊕H2 are considered. For an interval Δ which does not intersect the spectrum of at least one of the diagonal entries of A, we prove angular operator representations for the corresponding spectral subspace LΔ(A) of A and we study the supporting subspace in this angular operator representation of LΔ(A), which is the orthogonal projection of LΔ(A) to the corresponding component H1 or H2. Our main result is a description of a special direct complement of this supporting subspace in its component in terms of spectral subspaces of the values of the corresponding Schur complement of A in the endpoints of Δ.