Authors: Heinz W. Engl and M.Z. Nashed
Abstract: In 1956, R.Penrose studied best-approximate solutions of the matrix equation AX=B. He proved that A^+B (where A^+ is the Moore-Penrose inverse) is the unique matrix of minimal Frobenius norm among all matrices which minimize the Frobenius norm of AX-B. In particular, A^+ is the unique best-approximate solution of AX=I. The vector version of Penrose's result (that is, the fact that the vector A^+b is the best-approximate solution in the Euclidean norm of the vector equation Ax=b) has long been generalized to infinite dimensional Hilbert spaces. In this paper, an infinitie dimensional version of Penrose's full result is given. We show that a straightforward generalization is not possible and provide new extremal characterizations (in terms of Hermitian order) of A^+ and of the classes of generalized inverse assosciated with minimal norm solutions of consistent operator equations or with least-squares solutions. For a certain class of operators, we can phrase our characterizations in terms of a whole class of norms (including the Hilbert-Schmidt and the trace norms), thus providing new extremal characterizations even in the matrix case. We treat both operators with closed range and with non necessarily closed range. Finalla, we characterize A^+ as the unique inner inverse of minimal Hilbert-Schmidt norm if ||A^+||_2<\infty. We give an application of the new extremal characterization to the compensation problem in systems analysis in infinite-dimensional Hilbert spaces.