The property $ (\omega{ \pi }) $ as a generalization of the a-Weyl theorem

Wei Xu; Elvis Aponte; Ponraj Vasanthakumar; Wei Xu; Elvis Aponte; Ponraj Vasanthakumar

doi:10.3934/math.20241253

AIMS Mathematics

2024, Volume 9, Issue 9: 25646-25658. doi: 10.3934/math.20241253

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Research article

The property $ (\omega{ \pi }) $ as a generalization of the a-Weyl theorem

1.
Computer Science Department, New York University, 251 Mercer Street, New York, NY 10012, USA
2.
Departamento de Matemáticas, Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral, ESPOL, Campus Gustavo Galindo, km. 30.5 vía Perimetral, Guayaquil, 090902, Ecuador
3.
Department of Mathematics, Dr. N.G.P. Institute of Technology, Kalapatti Road, Coimbatore-641048, Tamilnadu, India

Received: 20 June 2024 Revised: 06 August 2024 Accepted: 15 August 2024 Published: 04 September 2024
MSC : Primary 47A10, 47A11; Secondary 47A53, 47A55

In this paper, for a bounded linear operator defined on a complex Banach space of infinite dimension, we consider the set of isolated points in its approximate point spectrum, which are eigenvalues of finite multiplicity; this set can be equal to the spectrum of the operator but without its upper semi-Fredholm spectrum, and this relation or equality defines in the literature a new spectral property called the property $ (\omega{ \pi }) $ and is a generalization of the classical a-Weyl theorem. We establish some characterizations and consequences about the property $ (\omega{ \pi }) $, some with topological aspects. Furthermore, we study this property through the Riesz functional calculus. Part of the spectral structure of a linear operator verifying property $ (\omega{ \pi }) $ is described, obtaining some associated properties.
- property $ (\omega{ \pi }) $,
- upper semi-Fredholm operator,
- SVEP
Citation: Wei Xu, Elvis Aponte, Ponraj Vasanthakumar. The property $ (\omega{ \pi }) $ as a generalization of the a-Weyl theorem[J]. AIMS Mathematics, 2024, 9(9): 25646-25658. doi: 10.3934/math.20241253

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Abstract

In this paper, for a bounded linear operator defined on a complex Banach space of infinite dimension, we consider the set of isolated points in its approximate point spectrum, which are eigenvalues of finite multiplicity; this set can be equal to the spectrum of the operator but without its upper semi-Fredholm spectrum, and this relation or equality defines in the literature a new spectral property called the property $ (\omega{ \pi }) $ and is a generalization of the classical a-Weyl theorem. We establish some characterizations and consequences about the property $ (\omega{ \pi }) $, some with topological aspects. Furthermore, we study this property through the Riesz functional calculus. Part of the spectral structure of a linear operator verifying property $ (\omega{ \pi }) $ is described, obtaining some associated properties.

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