Weighted weak group inverse for Hilbert space operators
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Weighted weak group inverse for Hilbert space operators ´ 1 , Daochang ZHANG2 Dijana MOSIC 1 Faculty of Sciences and Mathematics, University of Niˇs, P. O. Box 224, 18000 Niˇs, Serbia 2 College of Sciences, Northeast Electric Power University, Jilin 132012, China
c Higher Education Press 2020
Abstract We present the weighted weak group inverse, which is a new generalized inverse of operators between two Hilbert spaces, and we extend the notation of the weighted weak group inverse for rectangular matrices. Some characterizations and representations of the weighted weak group inverse are investigated. We also apply these results to define and study the weak group inverse for a Hilbert space operator. Using the weak group inverse, we define and characterize various binary relations. Keywords Weak group inverse, weighted core-EP inverse, Wg-Drazin inverse, Hilbert space MSC 47A62, 47A05, 15A09 1
Introduction
Throughout this paper, let B(X, Y ) be the set of all bounded linear operators from X to Y, where X and Y are infinite-dimensional complex Hilbert spaces. In the case that X = Y, we set B(X) = B(X, X). For A ∈ B(X, Y ), A∗ , N (A), R(A), and σ(A) represent the adjoint of A, the null space, the range, and the spectrum of A, respectively. We call P ∈ B(X) an idempotent if P 2 = P, and the orthogonal projector if P 2 = P = P ∗ . If L and M are closed subspaces, we denote by PL,M an idempotent on L along M, and by PL the orthogonal projector onto L. Let A ∈ B(X, Y )\{0}. There always exists B ∈ B(Y, X)\{0} such that BAB = B, which is not unique in general and it is called an outer inverse of A. The outer inverse is uniquely determined if we fix its range and kernel. For a subspace T of X and a subspace S of Y, the outer inverse B of A with the prescribed range T and the null space S is unique, if it exists, and denoted by (2) AT,S . We now present some special classes of outer inverses. Received April 10, 2020; accepted June 13, 2020 Corresponding author: Daochang ZHANG, E-mail: [email protected]
´ Daochang ZHANG Dijana MOSIC,
2
For a fixed operator W ∈ B(Y, X)\{0}, an operator A ∈ B(X, Y ) is called Wg-Drazin invertible [4] if there exists a unique operator B ∈ B(X, Y ) (denoted by Ad,w ) such that AW B = BW A,
BW AW B = B,
A − AW BW A is quasinilpotent.
If X = Y and W = I, then Ad = Ad,W is the generalized Drazin inverse (or the Koliha-Drazin inverse) of A [10]. We use B(X, Y )d,W and B(X)d , respectively, to denote the sets of all Wg-Drazin invertible operators in B(X, Y ) and generalized Drazin invertible operators in B(X). The W-weighted Drazin inverse is a particular case of the Wg-Drazin inverse for which A − AW BW A is nilpotent. The Drazin inverse is a special case of the generalized Drazin inverse for which A − A2 B is nilpotent, or equivalently, Ak+1 B = Ak for some non-negative integer k. The smallest such k is called the index of A and it is denoted by ind(A). In the case that ind(A) 6 1, A is group invertible and the group inverse A# of A is a special case of a Drazin inverse. The Drazin i
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