Common properties of bounded linear operators AC and BA: Spectral theory

Abstract

Let X, Y be Banach spaces, A:X→Y and B, C:Y→X be bounded linear operators satisfying the operator equation ABA=ACA. Recently, as extensions of Jacobson's lemma, Corach, Duggal and Harte studied common properties of AC−I and BA−I in algebraic viewpoint and also obtained some topological analogues. In this note, we continue to investigate common properties of AC and BA from the viewpoint of spectral theory. In particular, we give an affirmative answer to one question posed by Corach et al. by proving that AC−I has closed range if and only if BA−I has closed range.

Common properties of the operator equations in ultrametric specrtal theory

“…In particular, it is proved that N (I F − AS) is complemented in F if and only if N (I E − BA) is complemented in E. Moreover, the approach is generalized for considering relationships between the properties of I F − AS and I E − BD. We extend and develop some results of [1,2,11,12] in ultrametric spectral theory. Finally, several descriptive examples are furnished.…”

“…Recently, Corach et al [2] gave an extensions of Jacobson's lemma, they established some common properties of AS − I F and BA − I E when ABA = ASA. In [12], Zeng et Zhong continued to examine the joint properties of AS and BA from the attitude of classical spectral theory considering A ∈ B(E, F ) and B, S ∈ B(F, E) such that ABA = ASA, where E and F were assumed to be Banach spaces over C. In particular, they gave a positif answer to one question raised by the authors [2], by showing that AS − I F is of closed range if and only if BA − I E is of closed range. Yan and Fang [11] examined the joint properties of BD and AS in terms of regularity when A, D ∈ B(E, F ) and B, S ∈ B(F, E) satisfy the operator equations DBD = ASD and DBA = ASA.…”