Common properties of the operator products in spectral theory

Abstract

The concept of regularity was firstly introduced by Kordula and Müller. In this paper, we investigate the common properties of AC and BD in viewpoint of regularity when A, B, C and D all satisfy the operator equation set above.

New extensions of Cline’s formula for generalized inverses

“…But we also remark here that, for few operator properties, we have not yet find suitable analogues in this case (see [11,31]). Very recently, Yan and Fang [28,29] investigated a new extension of Jacobson's lemma and obtained many of its relatives in the case when…”

Generalized Cline's formula for G-Drazin inverse in a ring

“…By virtue of Theorem 2.5, AC ∈ L(X) D implies that BD ∈ L(X) D . This completes the proof by [11,Theorem 3.1].…”

“…Let a, b ∈ R. The Cline's formula for g-Drazin inverse stated that ab ∈ R d if and only if ba ∈ R d ad (ba) d = b[(ab) d ] 2 a (see [6,Theorem 2.1]). Cline's formula plays an important role in the generalized inverse of matrix and operator theory ( [2,4,7,9,11,12,13]). In [5,Theorem 2.3], Lian and Zeng proved the generalized Cline's formula to the case when aba = aca.…”

A note on the common spectral properties for bounded linear operators

“…The regularities R i , where 1 ≤ i ≤ 15, were introduced and studied in [10,12,14] but are in a different form. Regularity R 18 was introduced by [4], while R 16 , R 17 and R 19 were introduced by [18].…”

“…If C = I in the last condition, one can retrieve Schmoeger's result. For operators A, B, C and D satisfying ACD = DBD and BDA = ACA, Yan and Fang [17] investigated spectral properties for AC and BD. Recently, [5] studied common properties for ac and ba for elements in a ring satisfying a(ba) 2 = abaca = acaba = (ac) 2 a.…”