Spectral analysis of operator matrices: limit point insights

This paper explores the potential of local spectral theory to investigate the limit point set of the descent spectrum of upper triangular operator matrices, denoted by T , on Banach spaces. We rigorously prove that transitioning from the accumulation set of the diagonal descent spectrum, denoted by...

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Published in:Annali dell'Università di Ferrara. Sezione 7. Scienze matematiche Vol. 71; no. 1; p. 16
Main Author: Bahloul, Aymen
Format: Journal Article
Language:English
Published: Milan Springer Milan 01.03.2025
Springer Nature B.V
Subjects:
Algebraic Geometry
Analysis
Banach spaces
Geometry
History of Mathematical Sciences
Investigations
Mathematics
Mathematics and Statistics
Numerical Analysis
Operators (mathematics)
Spectral theory
Spectrum analysis
47A08
Upper triangular operator matrices
Limit points
Descent spectrum
47A11
Local spectral theory
Banach spaces
47A10
ISSN:0430-3202, 1827-1510
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Abstract This paper explores the potential of local spectral theory to investigate the limit point set of the descent spectrum of upper triangular operator matrices, denoted by T , on Banach spaces. We rigorously prove that transitioning from the accumulation set of the diagonal descent spectrum, denoted by Acc σ d ( T diag ) , to that of the complete descent spectrum, Acc σ d ( T ) , involves removing specific subsets within Acc σ d ( A 1 ) ∩ Acc σ a ( A 2 ) ∩ Acc σ a ( A 3 ) . Additionally, we present sufficient conditions that ensure the limit points of the descent spectrum of the operator matrix encompass the combined limit points of its diagonal entry spectra. This significantly addresses a longstanding question posed by Campbell (Linear Multilinear Algebra 14:195–198, 1983) regarding the limit points for the descent spectrum of the last 3 × 3 operator matrix form. Specifically, Campbell inquired about developing new methods to analyze the spectral properties of such matrices without resorting to partitioning their entries, a challenge that has remained unresolved for decades. Our findings provide a comprehensive solution, illustrating that a deeper understanding of the spectral behavior can be achieved by considering the entire matrix structure collectively.
AbstractList This paper explores the potential of local spectral theory to investigate the limit point set of the descent spectrum of upper triangular operator matrices, denoted by T, on Banach spaces. We rigorously prove that transitioning from the accumulation set of the diagonal descent spectrum, denoted by Accσd(Tdiag), to that of the complete descent spectrum, Accσd(T), involves removing specific subsets within Accσd(A1)∩Accσa(A2)∩Accσa(A3). Additionally, we present sufficient conditions that ensure the limit points of the descent spectrum of the operator matrix encompass the combined limit points of its diagonal entry spectra. This significantly addresses a longstanding question posed by Campbell (Linear Multilinear Algebra 14:195–198, 1983) regarding the limit points for the descent spectrum of the last 3×3 operator matrix form. Specifically, Campbell inquired about developing new methods to analyze the spectral properties of such matrices without resorting to partitioning their entries, a challenge that has remained unresolved for decades. Our findings provide a comprehensive solution, illustrating that a deeper understanding of the spectral behavior can be achieved by considering the entire matrix structure collectively.
This paper explores the potential of local spectral theory to investigate the limit point set of the descent spectrum of upper triangular operator matrices, denoted by T , on Banach spaces. We rigorously prove that transitioning from the accumulation set of the diagonal descent spectrum, denoted by Acc σ d ( T diag ) , to that of the complete descent spectrum, Acc σ d ( T ) , involves removing specific subsets within Acc σ d ( A 1 ) ∩ Acc σ a ( A 2 ) ∩ Acc σ a ( A 3 ) . Additionally, we present sufficient conditions that ensure the limit points of the descent spectrum of the operator matrix encompass the combined limit points of its diagonal entry spectra. This significantly addresses a longstanding question posed by Campbell (Linear Multilinear Algebra 14:195–198, 1983) regarding the limit points for the descent spectrum of the last 3 × 3 operator matrix form. Specifically, Campbell inquired about developing new methods to analyze the spectral properties of such matrices without resorting to partitioning their entries, a challenge that has remained unresolved for decades. Our findings provide a comprehensive solution, illustrating that a deeper understanding of the spectral behavior can be achieved by considering the entire matrix structure collectively.
ArticleNumber 16
Author Bahloul, Aymen
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Cites_doi 10.2478/mjpaa-2022-0024
10.1515/gmj-2022-2210
10.1007/s43036-024-00341-w
10.1016/j.bulsci.2024.103400
10.1080/01630563.2022.2137811
10.1007/s12215-019-00410-7
10.1080/03081088308817556
10.1007/BF01351564
10.1515/gmj-2024-2047
10.1093/oso/9780198523819.001.0001
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Copyright The Author(s) under exclusive license to Università degli Studi di Ferrara 2024 Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
The Author(s) under exclusive license to Università degli Studi di Ferrara 2024.
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Keywords 47A08
Upper triangular operator matrices
Limit points
Descent spectrum
47A11
Local spectral theory
Banach spaces
47A10
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KB Laursen (573_CR10) 2000
F Abdmouleh (573_CR3) 2023; 30
A Bahloul (573_CR6) 2022; 43
SL Campbell (573_CR9) 1983; 14
A Tajmouati (573_CR12) 2020; 69
M Burgos (573_CR8) 2006; 56
AE Taylor (573_CR13) 1980
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SubjectTerms Algebraic Geometry
Analysis
Banach spaces
Geometry
History of Mathematical Sciences
Investigations
Mathematics
Mathematics and Statistics
Numerical Analysis
Operators (mathematics)
Spectral theory
Spectrum analysis
Title Spectral analysis of operator matrices: limit point insights
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