Group, Moore-Penrose, core and dual core inverse in rings with involution

Abstract: Let $R$ be a ring with involution. The recently introduced notions of the core and dual core inverse are extended from matrix to an arbitrary $*$-ring case. It is shown that the group, Moore-Penrose, core and dual core inverse are closely related and they can be treated in the same manner using appropriate idempotents. The several characterizations of these inverses are given. Some new properties are obtained and some known results are generalized. A number of characterizations of EP elements in $R$ are obtained. It is shown that core and dual core inverse belong to the class of inverses along an element and to the class of $(b,c)$-inverses. The case when $R$ is algebra of all bounded linear operators on Hilbert space is specifically considered.