Matrices A such that AA† − A†A are nonsingular

In this paper we study the class of square matrices A such that AA† − A†A is nonsingular, where A† stands for the Moore-Penrose inverse of A. Among several characterizations we prove that for a matrix A of order n, the difference AA† − A†A is nonsingular if and only if R(A)⊕R(A∗)=Cn,1, where R(·) denotes the range space. Also we study matrices A such that R(A)⊥=R(A∗).