Banach triples with generalized inverses (Q1119724)

Penrose showed that for every complex rectangular matrix \(a\in M_{n\times m}({\mathbb{C}})\) there exists a unique \(b\in M_{n\times m}({\mathbb{C}})\), the generalized inverse of a, satisfying the following conditions: (i) \(aa^*b=a\), (ii) \(ba^*a=a\), (iii) \(bb^*a=b\), (iv) \(ab^*b=b\). These conditions can be expressed in terms of the triple product \(<abc>=ab^*c\) defined in \(A=M_{n\times m}({\mathbb{C}})\), \(b^*\) being the adjoint of b. With this triple product and with the operator norm, A is a complex Banach (associative) triple. In this note the authors determine all the complex Banach triples with generalized inverses.