Weak core inverses and pseudo core inverses in a ring with involution

On generalized core invertibility in Banach algebras

“…The g-Drazin inverse palys an important role in ring and matrix theory (see [1]). We note that the weak core inverse in [22] is a special case of generalized core inverse as the following shows.…”

“…We denote it by A W ❖, † . Recently, Many authors studied weak core inverse of complex matrices from different views, e.g., [9,11,12,16,17,22]. In [9], Fu et al considered the weak core inverse for complex square matrices.…”

“…In [9], Fu et al considered the weak core inverse for complex square matrices. In [22], Zhou and Chen studied the weak core inverse in a ring with proper involution. In [16], Mosić and Marovt investigated the weak core inverse for the bounded linear operator over a Hilbert space.…”

Pseudo core invertibility and DMP invertibility in two semigroups of a ring with involution

“…Additionally, Gao et al [8] discussed existence criteria and formulae of the pseudo core inverse of a companion matrix over a ring with involution. Zhou and Chen [27] showed some characterizations of pseudo core inverses and the relation with other generalized inverses. For more details of pseudo core inverses and DMP inverses, for example, see [18,28].…”