Common properties of the operator products in local spectral theory

The generalized inverses of the products of two elements in a ring

“…It has been realized that Jacobson's lemma has many suitable analogues for operator properties [1,2,16] and generalized inverses [5,7,20]. In [5,7], Patrício-Hartwig and Cvetković-Ilić-Harte proved that the Drazin invertibility of 1 − ac implies that of 1 − ca, respectively.…”

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…”

Common properties of the operator products in spectral theory

“…Moreover, it is known that R 1 and R 6 are the surjective and bounded below operators, respectively. Hence, the Proposition 2.2 and 2.5 in [ [12]] are contained in the following theorem. Proof.…”

“…In [[13]], Zeng and Zhong continued to investigate the common properties of AC and BA in the viewpoint of spectral theory. Recently, we gave a new extension of Jacobson's Lemma, in [ [12]]: If A, D ∈ B(X, Y ) and B, C ∈ B(Y, X) satisfy…”