Abstract
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.
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