AN APPROXIMATION THEOREM OF A M-P INVERSE BY OUTER INVERSES

摘要

Let H1 and H2 be separable Hilbert spaces, and B(H1, H2) all of bounded linear operators from H1 into H2. In this note, we prove the following theorem: for any positive integer N and T ∈ B(H1, H2) with a closed range, there exists an outer inverse T#N with finite rank N such that T+y = lira T#Ny for any y ∈ H2, where T+N →∞denotes the Moore-Penrose inverse of T. Thus computing T+ is reduced to computing outer inverses T#N with finite rank N. Moreover, because of the stability of bounded outer inverse of a T ∈ B(H1,H2), this is very useful.