Moore–Penrose-invertible normal and Hermitian elements in rings

On EP elements, normal elements and partial isometries in rings with involution

“…EP matrices and normal matrices, as well as EP linear operators and normal linear operators on Banach or Hilbert spaces have been investigated by many authors (see, for example, [1,2,3,4,5,7,8,9,10,11,13,15,17,18,20]). D. Mosić et al in [21,24] use the setting of rings with involution to investigate EP elements and normal elements, giving new characterizations to them and providing simpler and more transparent proofs to already existing ones. D. Mosić and D.S.…”

“…Chen R = M 2 (S) with the transposition involution * . Take a = [21]). Multiplying a from two sides of the equation (i), we get the equation(ii).…”

Weighted partial isometries and weighted-EP elements in C*-algebras

“…As a consequence, for e = f = 1, we obtain some well known characterizations of partial isometries, EP, star-dagger, normal and Hermitian elements. The identity (ab) * = b * a * are important when we proved the equivalent statements characterizing the condition of being a partial isometry, EP, star-dagger, normal and Hermitian element in a ring with involution R in [12,13]. Since ( * , e, f ): A → A is not in general an involution, in most statements an additional condition needs to be consider.…”

“…Various characterizations of MP-invertible normal and Hermitian elements in rings with involution were investigated in [13]. Some of these results were proved for complex square matrices in [3], using the rank of a matrix, or in [1], using an elegant representation of square matrices as the main technique.…”

Normal enveloping algebras

“…Over the decades, normal algebras with involutions have been extensively investigated on their own; see, for example, [Beidar et al 1981;Bovdi et al 1985;Bovdi 1990;1997;Bovdi and Siciliano 2007;Brešar and Vukman 1989;Herstein 1976;Knus et al 1998;Lim 1977;1979;Maxwell 1972]. Moreover, they have several applications in linear algebra and functional analysis; see, for example, [Berberian 1959;Fuglede 1950;Maxwell 1972;Mosić and Djordjević 2009;Putnam 1951;Yood 1974]. It is well-known that any normal algebra with involution satisfies the standard polynomial identity of degree 4 [Herstein 1976, Section 5].…”