Some Remarks on Perturbation Classes of Semi-Fredholm and Fredholm Operators

In this work, we present some results concerning the operators defined on various classes of exotic Banach spaces, containing in particular those studied respectively by V. Ferenczi [7,8] and T. Gowers with B. Maurey [14,15]. We show that, on hereditarily indecomposable or quotient hereditarily indecomposable Banach space X, the set of bounded Fredholm operators is dense in L(X), this gives that the boundary of bounded Fredholm operators is nothing else but the ideal of strictly singular operators if X is hereditarily indecomposable Banach space (resp. the ideal of strictly cosingular operators if X is quotient hereditarily indecomposable Banach space). On the other hand, a comparison between sufficiently rich and exotic Banach spaces is given via some properties of the two maps spectra and Wolf essential spectra.

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“…As it was already mentioned in [5], the sets F + (X) and F − (X) are closed sets, moreover, we have just the inclusions F + (X)…”

Section: Remark 25mentioning

confidence: 70%

“…We prove just the first assertion, the second one can be established in the same way. Let We present here a surprising result derived on the exotic Banach spaces and established by [5]. Theorem 2.5.…”

Section: Now Define the Operator A P Bymentioning

confidence: 89%

Some spectral properties of linear operators on exotic Banach spaces

Dehici

Debbar

2012

Lobachevskii J Math

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Properties (RB) and (gRB) for Bounded Linear Operators

Mehd

Mohsen²,

Fari³

2022

J. Phys.: Conf. Ser.

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In this article, we consider pseudo invertible operators for study of the relationship between the space of relatively regular operators and some generalizations of Weyl and Browder theorems. By using the analysis and representation of pseudo invertible operators, some new properties in connection with Browder’s type theorems, were presented for bounded linear operators T ∈ B(X). These properties, which we refer to as property (RB), imply that All poles of the resolvent of T of finite rank in the typical spectrum are precisely those places of the spectrum for which a reasonably regular operator with its pseudo inverse operator is surjective. (γI – T)† ∈ SU(X), In the usual spectrum, the set of all poles of the resolvent is exactly those points of the spectrum for which we call property (gRB). γI – T is a B-relatively regular operator with its pseudo inverse operator is surjective (γI – T)† ∈ SU(X). In addition, several sufficient and necessary conditions for which properties (RB) and (gRB) hold are given.

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Some Remarks on Perturbation Classes of Semi-Fredholm and Fredholm Operators

Cited by 2 publications

References 15 publications

Some spectral properties of linear operators on exotic Banach spaces

Some spectral properties of linear operators on exotic Banach spaces

Properties (RB) and (gRB) for Bounded Linear Operators

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