The Drazin Inverse Of Anti-triangular Operator Matrices

In this paper, the Drazin inverse and its representation of anti-triangular operatormatrixare studied by the method of the resolvent expansion and by the definition of the Drazin inverse, respectively.1) Under the condition of we use resolvent expansion method to obtain the existence and the representation of the Drazin inverse of the anti-triangular op-erator matrix M. We remark that the resolvent expression of the anti-triangular operator matrix M is the key of this method.2) Under the conditions of by using the definition of Drazin inverse, the existence and the representation of the Drazin inverse of the anti-triangular operator matrix M are presented. The key of this method is to find the index of the anti-triangular operator matrix M.The results above are extension of some recent researches under different conditions. Moreover, an example is presented to illustrate our results. However, the representation of Drazin inverse for a 2 x 2 block matrix still remains unsolved.