Characterizations and representations of the group inverse involving idempotents

Algebraic properties of the group inverse

“…However, in (3), instead of 1, the unit of the ring R, we have p at the corresponding position. This will have a huge impact on our work in a way that some of the analogues of theorems appearing in [3] will have proofs that no longer hold without some additional assumptions. For that reason, we will modify those assumptions so that they exclude some of the idempotents q represented as in (3) (this is different from [3] where we had corresponding results holding for all idempotents Q of the appropriate representation).…”

“…We shall study the group generalized inverse, which appears to have a central role in some applications [10,11,12,13], as well as in characterizations of some other generalized inverses [1, section 3.3]. Our main task will be to generalize some known results about bounded linear operators that appear in [3]. The question of the invertibility of p − q, p, q-idempotents, is especially important since it is connected with various other questions.…”

“…The question of the invertibility of p − q, p, q-idempotents, is especially important since it is connected with various other questions. For details, see introductory section in [3] and references mentioned there.…”

“…In [3] Deng has proved many interesting results concerning bounded linear idempotents P, Q ∈ B(H). Almost all of his proofs use the method of decomposing Hilbert space as H = R(P ) ⊕ N (P ) for some idempotent P , and then studying the matrix decomposition of certain operators with respect to this decomposition.…”

Generalized Drazin invertibility of combinations of idempotents

“…Consequently, this topic received attention in a number of papers, extending the ordinary invertibility to the Drazin invertibility, for instance [2,1,[4][5][6][7]13], and the difference and sum to a linear combination of idempotents [2,1,4,13]. These references by no means exhaust the literature on the topic.…”