Back to RS-SR spectral theory

Benhida, Chafiq; Zerouali, E. H.

doi:10.4064/bc75-0-4

Perspectives in Operator Theory 2007

DOI: 10.4064/bc75-0-4

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Back to RS-SR spectral theory

Chafiq Benhida

E. H. Zerouali²

Abstract: Abstract. Let X, Y be Banach spaces, S : X → Y and R : Y → X be bounded operators. We investigate common spectral properties of RS and SR. We then apply the result obtained to extensions, Aluthge transforms and upper triangular operator matrices.

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Cited by 3 publications

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New results on common properties of the products AC and BA, II

Zeng

Yan

Zhang

2020

Mathematische Nachrichten

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In this note, we continue to investigate common properties of the products in the setting of rings, bounded linear operators, or Banach algebras. We prove: (i) If are elements in a unital associative ring satisfying , then von Neumann regularity (resp. generalized Fredholmness relative to an ideal of ) of is converted into that of . (ii) If are bounded linear operators satisfying , then and share common complementability of kernels and ranges. (iii) If are elements in a unital semisimple Banach algebra satisfying and is a trace ideal of such that soc kh(soc, then and share common Fredholmness relative to and have the same abstract index. A similar result holds for B‐Fredholmness in primitive Banach algebra.

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New results on common properties of the products AC and BA, II

Zeng

Yan

Zhang

2020

Mathematische Nachrichten

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Back to the common spectral properties of operators satisfying AkBkAk = Ak+1 and BkAkBk = Bk+1

Hadji

Zguitti

2021

Asian-European J. Math.

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In [A. Gupta and A. Kumar, Common spectral properties of linear operators [Formula: see text] and [Formula: see text] satisfying [Formula: see text] and [Formula: see text], Asian-European J. Math. 12(5) (2019) 1950084], Gupta and Kumar studied the common spectral properties for two Banach bounded linear operators [Formula: see text] and [Formula: see text] satisfying [Formula: see text] and [Formula: see text] for some positive integer [Formula: see text]. In this note, we study the common local spectral properties for such operators by showing that [Formula: see text] and [Formula: see text] share the single valued extension property, the Bishop property [Formula: see text] the property [Formula: see text] the decomposition property [Formula: see text] and decomposability. The closedness of analytic core and the quasinilpotent part and then the Dunford’s property [Formula: see text] and the property [Formula: see text] are also investigated.

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Common local spectral properties for the operator products

Hadji

2023

Asian-European J. Math.

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In their paper [Generalized Clines Formula for g-Drazin Inverse in a Ring, Filomat 35(8) (2021) 2573–2583], Chen and Abdolyousefi examined the common spectral properties of operators [Formula: see text] and [Formula: see text], subject to the following conditions: [Formula: see text] [Formula: see text] In this note, we explore the common local spectral properties of these operators. We demonstrate that [Formula: see text] and [Formula: see text] shared the single-valued extension property, the Bishop property [Formula: see text], the property [Formula: see text], the decomposition property [Formula: see text], and decomposability. Furthermore, we investigate the closedness of the analytic core and the quasinilpotent part. We also delve into the Dunford’s property [Formula: see text] and the property [Formula: see text]

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Back to RS-SR spectral theory

Abstract: Abstract. Let X, Y be Banach spaces, S : X → Y and R : Y → X be bounded operators. We investigate common spectral properties of RS and SR. We then apply the result obtained to extensions, Aluthge transforms and upper triangular operator matrices.

Cited by 3 publications

References 12 publications

New results on common properties of the products AC and BA, II

New results on common properties of the products AC and BA, II

Back to the common spectral properties of operators satisfying AkBkAk = Ak+1 and BkAkBk = Bk+1

Common local spectral properties for the operator products

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