We investigate the determinantal representation by exploiting the limiting expression for the generalized inverseAT,S(2). We show the equivalent relationship between the existence and limiting expression ofAT,S(2)and some limiting processes of matrices and deduce the new determinantal representations ofAT,S(2), based on some analog of the classical adjoint matrix. Using the analog of the classical adjoint matrix, we present Cramer rules for the restricted matrix equationAXB=D, R(X)⊂T, N(X)⊃S∼.
In this paper, we construct a new iterative method for computing the Drazin inverse and deduce the necessary and sufficient condition for its convergence to A d. Moreover, we present the error bounds of the iterative methods for approximating A d .
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| ...
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