Pseudo-inverses of operators
J. J. Koliha · Bulletin of the American Mathematical Society · 1974
1. Let X and Y be complex Banach spaces, A a bounded linear operator from X to Y. If the null space N(A) and the closed range R(A)~ possess closed complementary subspaces U in X and V in Y respectively, the pseudo-inverse A* of A relative to (U, V) is defined as the linear extension of (AlU)to D(A*)=R(A)+V With the null space N(A*)=V. (This is a generalization to Banach space of the standard pseudo-inverse of a Hubert space operator (cf. [8]). If R(A) is closed, the definition agrees with the ones given in [1] and [7]. In this case A is defined and bounded on all of Y.) If U==R(B)~ and V=N(B) for some bounded linear operator B: Y-+X, A will be called the pseudo-inverse of A relative to B, written A'. Proposition 6 of [6] leads to the following result.