Note on the iterative methods for computing the generalized inverse over Banach spaces

Abstract

In this paper, we construct iterative methods for computing the generalized inverse A (2) T,S over Banach spaces, and also for computing the generalized Drazin inverses a d of Banach algebra element a. Moreover, we estimate the error bounds of the iterative methods for approximating A (2) T,S or a d .generalized Drazin inverse a d of Banach algebra element a. In Section 4, we give an example for computing A (2) T,S to exploit those new iterations. In this paper, we always assume that any Banach space is over the complex number field C and its dimension is not zero. Let X and Y be arbitrary Banach spaces. Then the symbol B(X, Y) denotes the set of all bounded linear operators from X to Y, in particular, B(X) := B(X, X), a complex Banach algebra with unit 1. For any A ∈ B(X, Y), we denote its range, null space and norm by R(A), N(A) and A , respectively. Further, we say that A is regular if there exists an X ∈ B(Y, X) such that AXA = A and that A has a {2} (or outer) inverse if there exists an X ∈ B(Y, X) such that XAX = X . If A ∈ B(X), then we denote its spectrum and spectral radius by (A) and (A), respectively.Let L, M ⊂ X with L⊕M = X. Then the symbol P L,M stands for an operator that is called a projection from X onto L if it is a bounded linear map from X onto L and P 2 L,M = P L,M . It is well known that a closed subspace L of a Banach space X is complemented in X if and only if there exists a projection from X onto L.Let H i , i = 1, 2, be Hilbert spaces and A ∈ B(H 1 , H 2 ). Recall that the Moore-Penrose inverse of A is defined as the unique operator A † ∈ B(H 2 , H 1 ) satisfying the Penrose equations [1, 7, 11] And it is well known thatThroughout, let A be a complex Banach algebra with the unit 1. The symbols ann l (a) and ann r (a), respectively, stand for the left and right annihilators of a in A. Let p ∈ A be idempotent. Then pA p = { pap : a ∈ A} is a subalgebra of A with unit p. Thus, for a ∈ A, if there exists an element b ∈ pA p such that ab = ba = p, then we say that a is invertible in pA p and b is denoted by a| −1 pA p . Recall that an element b ∈ A is the generalized Drazin inverse of a (or Koliha-Drazin inverse of a), if the following hold: bab = b, ab = ba, and a(1−ab) is quasinilpotent.If the generalized Drazin inverse of a exists, then it is denoted by a d (see [15] for more details). In particular, if b = a d and a(1−ab) = 0, then b is called the group inverse of a and is denoted by a g .The following lemmas are needed in what follows. Lemma 1.1 (Müller [16, Chapter 1]) Let a ∈ A. Then (i) (a) is a nonempty closed subset of C. (ii) (Spectral Mapping Theorem for Polynomials) if f is a polynomial, then ( f (a)) = f ( (a)). (iii) lim n→∞ a n = 0 if and only if (a)<1. Lemma 1.2 (Djordjević and Stanimirović [17, Section 4]) Let X and Y be Banach spaces, A ∈ B(X, Y), T and S, respectively, be closed subspaces of X and Y. Then the following statements are equivalent: (i) A has a {2} inverse B ∈ B(Y, X) such that R(B) = T and N(B) = S; (ii) T is a complemented subspace of X, A(T ) is closed, A| ...

A hyperpower iterative method for computing the generalized Drazin inverse of Banach algebra element

“…1. An iterative method faster than (1) was developed in [23]. A family of hyperpower iterative procedures was investigated in [24].…”