On the mixed-type generalized inverses of the products of two operators

Abstract

Let $A$, $B$ and $AB$ be closed range operators. The explicit matrix expressions for various generalized inverses are obtained by using block operator matrix methods. Some subtle relationships  between the properties of  sub-blocks in operator matrices $A$, $B$ and their range  relations are built. New necessary and sufficient conditions for the equivalent relations, inclusion relations  and mixed-type generalized inverses relations are presented. Some recent mixed-type reverse-order laws results are covered and many new mixed-type generalized inverses relations are established by using this block-operator matrix technique.