Consistent invertibility and Weyl's theorem

Abstract

A Banach space operator T ∈ B( X) may be said to be "consistent in invertibility" provided that for each S ∈ B( X), T S and S T are either both or neither invertible. The induced spectrum contributes the conditions equivalent to various forms of "Weyl's theorem".

Jacobson's lemma for the generalized \(n\)-strong Drazin inverses in rings and in operator algebras

“…Djordjevic ( [16]) further described the CI operators on a Banach space. In [5], Cao et al investigated the CI spectrum on a Banach space, and applied the CI spectrum to characterize those operators T satisfying Weyl type theorems. The relation between CI spectrum and Weyl type theorems also can be seen in [27,28].…”

A Note on Operators Consistent in Invertibility

“…It is easy to see that an operator T ∈ B(H) is not CI if and only if T is left invertible but not right invertible, or right invertible but not left invertible. The CI spectrum of T ∈ B(H) is defined by σ CI (T ) = {λ ∈ C : T − λI is not CI} Results concerning CI operators were obtained in [8,9] and [1,2,10]. It is fairly easy to see that if A and B are CI operators, then AB is a CI operator, it would be of interest to determine whether the set of all CI operators is a regularity.…”

Left-right consistency in rings III

“…We remark that, in a change from [8], we are following the notation of Cao/Zhang/ Zhang [1]. The characterization [4] of left-right consistent elements can ([7] Theorem 4) be expressed spectrally:…”