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For any singular matrix A ∈ C n × n , let AD, A π = I n − A A D be its Drazin inverse and spectral projector, respectively. Let | | · | | be any norm on C n × n and A ¯ ∈ C n × n be any stable perturbation of A. In this note a sufficient condition is given under which | | A ¯ π − A π | | < 1. [ABSTRACT FROM AUTHOR]

Let be the set of bounded linear operators on a Hilbert space H, and E be an anti-triangular operator matrix defined by where 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. [ABSTRACT FROM AUTHOR]

The general exact solutions of the second-order homogeneous algebraic differential equation(Section.Display) is studied, whereandare three known bounded linear operators on a Banach space. Ifsuch thatis invertible, an anti-triangular operator matrixis defined. Whenis Drazin invertible, a necessary and sufficient condition is obtained in terms ofand its Drazin inverseunder whichbecomes a solution of (0.1). Specifically, a formula foris derived under the assumptions thatis a Hilbert space,andare positive semi-definite operators onwith the summationbeing positive definite. [ABSTRACT FROM PUBLISHER]

The second-order homogeneous algebraic differential equationis studied, whereare known such thatis non-singular for some. A formula for the general exact solutions of the above equation is provided under the condition that there exists, such that. For a special second-order homogeneous algebraic differential equation coming from a physics model investigated by Bhat and Bernstein, a detailed description is given about how to construct a particular solutionof the latter quadratic matrix equation. [ABSTRACT FROM AUTHOR]

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]

Abstract: Let be the set of bounded linear operators on a Banach space X, and be Drazin invertible. An element is said to be a stable perturbation of A if B is Drazin invertible and is invertible, where I is the identity operator on X, and are the spectral projectors of A and B, respectively. Under the condition that B is a stable perturbation of A, a formula for the Drazin inverse is derived. Based on this formula, a new approach is provided to the computation of the explicit Drazin indices of certain operator matrices. [Copyright &y& Elsevier]