Additive results of the Drazin inverse for bounded linear operators
Let \(\mathcal X\) be a Banach space and \(A\) be a bounded linear operator on \(\mathcal X\). If there exists a bounded linear operator \(T\) on \(\mathcal X\) such that \(AT=TA\), \(AT^2=T\) and \(A^{k+1}T=A^k\), where \(k\) is the index of \(A\), then such a \(T\) is called the Drazin inverse of \(A\). This generalized inverse is unique if it exists and is denoted by \(A^D\). However, it may not exist. There is a well known result of \textit{M. P. Drazin} [Am. Math. Mon. 65, 506--514 (1958; Zbl 0083.02901)] for the Drazin inverse of a sum of two operators, viz., \((P+Q)^D=P^D+Q^D\) whenever \(P\) and \(Q\) are Drazin invertible operators satisfying the conditions \(PQ=QP=0\). The authors of the paper under review show that \(P+Q\) is Drazin invertible if \(P^2Q+PQ^2=0\), \(P^3Q=0=PQ^3\) and \(PQ\) is Drazin invertible. In such a case, a rather complicated representation is proved for the Drazin inverse of \(P+Q\).
scientific article; zbMATH DE number 6336134