Reverse order laws in C∗-algebras

On Weighted Reverse Order Laws for the Moore–Penrose Inverse andK-Inverses

“…Under the conditions of the theorem, using the matrix representations given in Remark 3.1, it is not difficult to prove that b † a † c = b † a † and that necessary and sufficient condition for (ii) to holds is the fact that b{1, 3} · a{1, 3} ⊆ (ab){1, 3}. In particular, it is enough to prove the equivalences among statements (i)-(iv) for the case c = e. Now, the proof of this case follows by Theorem 3.1 in Cvetković-Ilić, Harte (2011), where the same conditions of statements (i)-(iv) were considered for a, b two C * -algebra elements and c = e. However, for the sake of completeness the proof of the case c = e will be presented.…”

Reverse-order law for the Moore–Penrose inverses of tensors

“…Hartwig [11] provided necessary and sufficient conditions for holding of triple (or three term) reverse order law (i.e., (ABC) † = C † B † A † where A, B and C are matrices). The study of this problem for generalized inverses in C*-algebras can be seen in the work by Cvetkovic-Ilic and Hartee [7] and Mosic and Djordjevic [17]. While Deng [8] studied the same problem for the group invertible operators, Wang et al [22] considered for the Drazin invertible operators.…”

Some relations of projection and star order in Hilbert space

“…Here, S ∈ B(H) is called the Moore-Penrose inverse of T , denoted by S = T + . It is well known that T has the MP-inverse if and only if R(T ) is closed, and the MP-inverse of T is unique (see [3][4][5], [8]). …”