Continuity and differentiability of the Moore-Penrose inverse in $C^*$-algebras
J. J. Koliha · MATHEMATICA SCANDINAVICA · 2001
The paper gives an elementary proof of the theorem on the continuity of the Moore^Penrose inverse in a C -algebra that does not require the concept of the conorm, but uses instead a C modification of Izumino's inequality kbyk 4kayk valid when a; b have the Moore^Penrose inverse and satisfy the inequalities kby ak < 2 kayky1 and kbby y aayk < 1. The paper then studies the conditions for the differentiability of the Moore^Penrose inverse in a C algebra and gives an explicit formula for the derivative. The Moore^Penrose inverse of an element a of a unital C -algebra A with the unit e is the unique element ay of A satisfying the equations aaya a; ayaay ay; aya aya; aay aay 1:1 (see [10, 5, 11, 13]). The set of all a 2 A that possess the Moore^Penrose inverse will be denoted by Ay. It is shown in [5, Theorem 6] that a 2 Ay if and only if a 2 aAa. The elements aya and aay are Hermitian idempotents. We also write Ay1 for the set of all invertible elements in A. It is well known that the following two results hold for the ordinary inverse in Banach algebras.