Representations for the Drazin inverse of an anti-triangular block operator matrix E with ind(E) ≤ 2
Qingxiang Xu, Chuanning Song, Lili He · Linear and Multilinear Algebra · 2017
Let be the set of bounded linear operators on a Hilbert space H, and E be an anti-triangular operator matrix defined bywhere and I is the identity operator on H. Two associated operators are introduced, and it is proved that if and only if is invertible for any . A new kind of anti-triangular operator matrices is constructed such that the Drazin index of each member is exactly less than or equal to 2. Thus a generalization of He et al. [General exact solutions of certain second-order homogeneous algebraic differential equations. Linear Multilinear Algebra. 64;2016:1011–1031] is obtained.