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Let A be a unital C*-algebra, a, x and y are elements in A. In this paper, we present the expression of the Moore-Penrose inverse and the group inverse of a-xy* under the conditions x = aa+ x, y* = y* a+ a, respectively. [ABSTRACT FROM AUTHOR]

Let X, Y be Banach spaces and T : X → Y be a bounded linear operator. In this paper, we initiate the study of the perturbation problems for bounded homogeneous generalized inverse Th and quasi-linear projector generalized inverse TH of T. Some applications to the representations and perturbations of the Moore-Penrose metric generalized inverse TM of T are also given. The obtained results in this paper extend some well-known results for linear operator generalized inverses in this ?eld. [ABSTRACT FROM AUTHOR]

Let L be a subspace of ${\bf R}^n$ and P L be the orthogonal projection of R n onto L . Then for the n × n matrix A , the generalized Bott-Duffin (B-D) inverse $A^{(+)}_{(L)}$ is given by $A^{(+)}_{(L)}=P_L(AP_L+I-P_L)^+$ . In this article we prove that $A^{(+)}_{(L)}=(P_LAP_L)^+$ iff $AL\cap L^\perp =0$ . This result extends the concept so-called the " L -s.p.d" matrix proposed by Chen in his article "The Generalized Bott-Duffin Inverse and its Applications". In the rest part of the article, the perturbation analysis of $A^{(+)}_{(L)}$ and the least squares solution of the systems $Ax+B^*y=b, Bx=d$ are established under certain small perturbation of A . [ABSTRACT FROM AUTHOR]

We show that if Α has tracial topological rank no more than k, then U[subm](A)/U[subm,sup0] (Α) ≅ K[sub1] (Α) for m ≥ 1+ *[These characters cannot be converted to ASCII text] and if p, q ∈ A are two projections with [p] = [q] in K[sub0](Α), then diag(p, 1[subm]) ∼ diag(g, 1[subm]) where m = max {1, *[These characters cannot be converted to ASCII text]}. Other results for K[sub0](A) are also established. [ABSTRACT FROM AUTHOR]