TAILIEUCHUNG - The Moore-Penrose inverse of differences and products of projectors in a ring with involution

In this paper, we study the Moore-Penrose inverses of differences and products of projectors in a ring with involution. Some necessary and sufficient conditions for the existence of the Moore-Penrose inverse are given. Moreover, the expressions of the Moore-Penrose inverses of differences and products of projectors are presented. | Turk J Math (2016) 40: 1316 – 1324 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article The Moore–Penrose inverse of differences and products of projectors in a ring with involution Huihui ZHU1,2 , Jianlong CHEN1,∗, Pedro PATR´ ICIO2,3 Department of Mathematics, Southeast University, Nanjing, . China 2 Center of Mathematics (CMAT), University of Minho, Braga, Portugal 3 Departament of Mathematics and Applications, University of Minho, Braga, Portugal 1 Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we study the Moore–Penrose inverses of differences and products of projectors in a ring with involution. Some necessary and sufficient conditions for the existence of the Moore–Penrose inverse are given. Moreover, the expressions of the Moore–Penrose inverses of differences and products of projectors are presented. Key words: Moore–Penrose inverses, normal elements, involutions, projectors 1. Introduction Throughout this paper, R is a unital ∗ -ring, that is a ring with unity 1 and an involution a 7→ a∗ satisfying that (a∗ )∗ = a , (a + b)∗ = a∗ + b∗ , (ab)∗ = b∗ a∗ . Recall that an element a ∈ R is said to have a Moore–Penrose inverse (abbr. MP-inverse) if there exists b ∈ R such that the following equations hold [11]: aba = a, bab = b, (ab)∗ = ab , (ba)∗ = ba. Any b that satisfies the equations above is called a MP-inverse of a. The MP-inverse of a ∈ R is unique if it exists and is denoted by a† . By R† we denote the set of all MP-invertible elements in R . MP-inverse of differences and products of projectors in various sets attracts wide attention from many scholars. For instance, Cheng and Tian [1] studied the MP-inverses of pq and p − q , where p , q are projectors in complex matrices. Li [10] investigated how to express MP-inverses of product pq and differences p − q and pq − qp , for two given projectors p and q in a C ∗

• Toán học
• Moore–Penrose inverses
• Normal elements
• Sufficient conditions
• MP invertible elements
• Purely ring theoretical method