The DMP Invertibility Of The Anti-Triangular Operator Matrix

Since the concept of the generalized inverse is introduced,it has been considerably investigated by scholars.The generalized inverse has wide applications in many areas such as operator theory,numerical analysis,perturbation eigenvalue,cryptography and Markov chain.Some new generalized inverses are introduced during research,for example,core inverse,CMP inverse,DMP inverse,etc.For the DMP inverse,scholars have studied the equivalent definition,shape,prop-erties,maximal classes,representations and iterative calculation.In this paper,we dis-cuss the DMP invertibility of the anti-triangular operator matrix(?)and the representation of DMP inverse in the Hilbert space,and we also study the Drazin invertibility of a class of four-block operator matrices(?).The main contents are as follows:1.For the anti-triangular operator matrix MI.Firstly,the solvability of the op-erator equation AX+X2=B is discussed,and invertible solutions of the equation are obtained under conditions(1)B2A=AB,(2)A2B=BA,(3)BA=A-1B,respectively.Secondly,by the space decomposition method,the relationship is es-tablished between the Drazin invertibility of MI and the solvability of the equation AX+X2=B,then the Drazin invertibility of MI is proved,and representations of Drazin inverse are obtained.Finally,combined with the MP invertibility,the DMP invertibility of MI and its expression are obtained.2.For the operator matrix M,based on the above method,the Drazin invertibility of MI is investigated under more general conditions.