Representations Of The Drazin Inverse Of Certain Two By Two Block Operator Matrices

As a very important mathematical object, Drazin inverses of block matrices have ap-plications in differential-algebraic equations, Markov chains, the control theory and some other scientific areas. The Drazin inverse of a block operator matrix is a natural general-ization of the generalized inverse of matrix, which is of great importance both theoretically and practically.Let Cm×n be the set of all m x n complex matrices. C(n+m)×(n+m) be an anti-triangular matrix, where A ∈ Cn×n. Let B+ be the Moore-Penrose inverse of B and I be the identity matrix. Under the condition of rank(BCB)= rank(B) and (I-BB+)AB= 0, (0.3) C. Bu, L. Sun, J. Zhou and Y. Wei [Some results on the Drazin inverse of anti-triangular matrices, Linear Multilinear Algebra 61 (2013),1568-1576] use the method of the matrix SVD, and get a concrete expression of the Drazin inverse of M. The purpose of this paper is to generalize the result above from the finite case to the infinite case.For any Hilbert spaces H1 and H2, let B(H1, H2) be the set of all bounded linear operators from H1 to H2. For any A ∈ B(H1, H2),R(A) and N(A) represent the range and the kernel of A, respectively. Let be an anti-triangular operator matrix, where A ∈ B(H1, H1) and I is the identity operator on H1. Suppose B is Moore-Penrose invertible and (I-BB+)A(I-BB+) is Drazin invertible. Under the condition of(I-BB+)AB= 0,N(BCBB+)= N(BB+) and R(BCBB+)= R(BB+), (0.4) this paper gives a concrete expression of the Drazin inverse of E. When both H1 and H2 are finite spaces, this paper proves the conditions (0.3) and (0.4) are the same. Thus, a generalization of the result of C Bu, L. Sun, J. Zhou and Y. Wei is obtained.