Representations for the group inverse of anti-triangular block operator matrices

For any Hilbert spaces H1H1 and H2H2, let B(H1,H2)B(H1,H2) be the set of bounded linear operators from H1H1 to H2H2. In this paper, necessary and sufficient conditions are given under which an anti-triangular block operator matrix E=(ABC0) is group invertible, where A∈B(H1,H1),B∈B(H2,H1)A∈B(H1,H1),B∈B(H2,H1) and C∈B(H1,H2)C∈B(H1,H2). In the case that E is group invertible, a new formula for the group inverse of E is derived under the only restriction that certain associated operators have closed ranges. This gives especially a new characterization of the group inverse of an anti-triangular block matrix without restrictions on its individual block matrices.