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

Abstract: For any Hilbert spaces and , let be the set of bounded linear operators from to . In this paper, necessary and sufficient conditions are given under which an anti-triangular block operator matrix is group invertible, where and . 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. [Copyright &y& Elsevier]