Computational Models for <i>(M, K)</i>-Quasi-*-Parahyponormal Operators

Kiratu Beth; Ngoci Abishag; Obiero Ben

doi:doi:10.11648/j.pamj.20251405.13

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Computational Models for (M, K)-Quasi-*-Parahyponormal Operators

Kiratu Beth^*

, Ngoci Abishag

, Obiero Ben

Published in Pure and Applied Mathematics Journal (Volume 14, Issue 5)

Received: 7 August 2025 Accepted: 25 August 2025 Published: 25 September 2025

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Abstract

We study the two-parameter class of (M, k)-Quasi-∗-Parahyponormal operators on separable Hilbert spaces, which strictly enlarges the traditional parahyponormal and paranormal hierarchies. Analytically we prove three fundamental results: Every operator in the class has finite ascent and enjoys the single-valued extension property (SVEP); The Browder–Weyl partition holds, so Weyl’s theorem is valid; A non-trivial closed invariant subspace exists whenever the commutant contains a non-zero compact element. Complementing these theorems, we introduce a proposed computational framework that realises the abstract operators as large weighted-shift matrices, verifies the defining quadratic inequality, and computes eigenvalues as well aaccelerated pseudospectra. Together, the analytic results and the computational framework deepen the spectral theory of (M, k)-Quasi-∗-Parahyponormal Operators and supply the first large-scale numerical evidence for their structural properties.

Published in	Pure and Applied Mathematics Journal (Volume 14, Issue 5)
DOI	10.11648/j.pamj.20251405.13
Page(s)	120-129
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Local Spectral Theory, SVEP, Weyl’s Theorem, Invariant Subspaces, Weighted-shift Models

References

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[2]	L. A. Coburn, “Weyl’s theorem for nonnormal operators,” Michigan Mathematical Journal, vol. 13, no. 3, pp. 285-288, 1966. https://doi.org/10.1307/mmj/1031732778
[3]	R. E. Curto and Y. M. Han, “Weyl’s theorem, a-Weyl’s theorem, andlocalspectraltheory,” JournaloftheLondon Mathematical Society, vol. 67, no. 2, pp. 499-509, 2003. https://doi.org/10.1112/S0024610702004027
[4]	P. Aiena, Fredholm and local spectral theory, with applications to multipliers. Springer Science & Business Media, 2004. https://doi.org/10.1007/1-4020-2525-4
[5]	M. H. M. Rashid, M. S. M. Noorani, and A. S. Saari, “Weyl’s type theorems for quasi-class A operators,” Journal of Mathematics and Statistics, vol. 4, no. 2, p. 70, 2008. https://doi.org/10.3844/jmssp.2008.70.74
[6]	A. F. Ariouat, A. N. Bakir, and A. Benali, “Bishop’s property, Weyl’s theorem and Riesz idempotent,” Journal of Computational Analysis and Applications, vol. 34, no. 1, 2025. https://eudoxuspress.com/index.php/pub/article/view/1788
[7]	A. N. Bakir, “A large class extending *-parahyponormal operators,” Journal of Science & Arts, vol. 23, no. 2, 2023. https://doi.org/10.46939/jscarts.23.2.001
[8]	N. Aronszajn and K. T. Smith, “Invariant subspaces of completely continuous operators,” Annals of Mathematics, vol. 60, no. 2, pp. 345-350, 1954. https://doi.org/10.2307/1969708
[9]	A. R. Bernstein, Invariant subspaces of polynomially compact operators on Banach space, Ph.D. thesis, 1967. https://msp.org/pjm/1967/21-3/pjm-v21-n3-p05-s.pdf
[10]	V. I. Lomonosov, “Invariant subspaces for the family of operators which commute with a completely continuous operator,” Functional Analysis and Its Applications, vol. 7, no. 3, pp. 213-214, 1973. https://doi.org/10.1007/BF01075854
[11]	S. W. Brown, “Hyponormal operators with thick spectra have invariant subspaces,” Annals of Mathematics, vol. 125, no. 1, pp. 93-103, 1987. https://doi.org/10.2307/1971301
[12]	J. Eschmeier and B. Prunaru, “Invariant subspaces and localizable spectrum,” Integral Equations and Operator Theory, vol. 42, pp. 461-471, 2002. https://doi.org/10.1007/BF01270923
[13]	A. N. Bakir, “Local spectral properties of (m,k)-quasi-*-paranormal operators,” Journal of Operator Theory, vol. 90, no. 1, pp. 123-147, 2023. https://doi.org/10.7900/jot.2021dec14.2388
[14]	V. I. Lomonosov, “Invariant subspaces for operators commuting with compact operators,” Functional Analysis and Its Applications, vol. 7, no. 3, pp. 213-214, 1973. https://doi.org/10.1007/BF01080698
[15]	K. B. Laursen and M. Neumann, An introduction to local spectral theory. Oxford University Press, 2000. Available: https://isbnsearch.org/isbn/9780198523819
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APA Style

Beth, K., Abishag, N., Ben, O. (2025). Computational Models for (M, K)-Quasi-*-Parahyponormal Operators. Pure and Applied Mathematics Journal, 14(5), 120-129. https://doi.org/10.11648/j.pamj.20251405.13

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ACS Style

Beth, K.; Abishag, N.; Ben, O. Computational Models for (M, K)-Quasi-*-Parahyponormal Operators. Pure Appl. Math. J. 2025, 14(5), 120-129. doi: 10.11648/j.pamj.20251405.13

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AMA Style

Beth K, Abishag N, Ben O. Computational Models for (M, K)-Quasi-*-Parahyponormal Operators. Pure Appl Math J. 2025;14(5):120-129. doi: 10.11648/j.pamj.20251405.13

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@article{10.11648/j.pamj.20251405.13,
  author = {Kiratu Beth and Ngoci Abishag and Obiero Ben},
  title = {Computational Models for (M, K)-Quasi-*-Parahyponormal Operators
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {14},
  number = {5},
  pages = {120-129},
  doi = {10.11648/j.pamj.20251405.13},
  url = {https://doi.org/10.11648/j.pamj.20251405.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20251405.13},
  abstract = {We study the two-parameter class of (M, k)-Quasi-∗-Parahyponormal operators on separable Hilbert spaces, which strictly enlarges the traditional parahyponormal and paranormal hierarchies. Analytically we prove three fundamental results: Every operator in the class has finite ascent and enjoys the single-valued extension property (SVEP); The Browder–Weyl partition holds, so Weyl’s theorem is valid; A non-trivial closed invariant subspace exists whenever the commutant contains a non-zero compact element. Complementing these theorems, we introduce a proposed computational framework that realises the abstract operators as large weighted-shift matrices, verifies the defining quadratic inequality, and computes eigenvalues as well aaccelerated pseudospectra. Together, the analytic results and the computational framework deepen the spectral theory of (M, k)-Quasi-∗-Parahyponormal Operators and supply the first large-scale numerical evidence for their structural properties.
},
 year = {2025}
}

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TY  - JOUR
T1  - Computational Models for (M, K)-Quasi-*-Parahyponormal Operators

AU  - Kiratu Beth
AU  - Ngoci Abishag
AU  - Obiero Ben
Y1  - 2025/09/25
PY  - 2025
N1  - https://doi.org/10.11648/j.pamj.20251405.13
DO  - 10.11648/j.pamj.20251405.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 120
EP  - 129
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20251405.13
AB  - We study the two-parameter class of (M, k)-Quasi-∗-Parahyponormal operators on separable Hilbert spaces, which strictly enlarges the traditional parahyponormal and paranormal hierarchies. Analytically we prove three fundamental results: Every operator in the class has finite ascent and enjoys the single-valued extension property (SVEP); The Browder–Weyl partition holds, so Weyl’s theorem is valid; A non-trivial closed invariant subspace exists whenever the commutant contains a non-zero compact element. Complementing these theorems, we introduce a proposed computational framework that realises the abstract operators as large weighted-shift matrices, verifies the defining quadratic inequality, and computes eigenvalues as well aaccelerated pseudospectra. Together, the analytic results and the computational framework deepen the spectral theory of (M, k)-Quasi-∗-Parahyponormal Operators and supply the first large-scale numerical evidence for their structural properties.

VL  - 14
IS  - 5
ER  -

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Author Information

Kiratu Beth

Department of Mathematics and Statistics, School of Science and Technology, Open University of Kenya, Nairobi, Kenya

Contact Email

http://orcid.org/0000-0003-2351-4285
Ngoci Abishag

Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Technical University of Kenya, Nairobi, Kenya

Contact Email

http://orcid.org/0009-0005-4337-1279
Obiero Ben

Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Technical University of Kenya, Nairobi, Kenya

http://orcid.org/0009-0002-0345-9928

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Plain Text BibTeX RIS

APA Style

Beth, K., Abishag, N., Ben, O. (2025). Computational Models for (M, K)-Quasi-*-Parahyponormal Operators. Pure and Applied Mathematics Journal, 14(5), 120-129. https://doi.org/10.11648/j.pamj.20251405.13

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ACS Style

Beth, K.; Abishag, N.; Ben, O. Computational Models for (M, K)-Quasi-*-Parahyponormal Operators. Pure Appl. Math. J. 2025, 14(5), 120-129. doi: 10.11648/j.pamj.20251405.13

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AMA Style

Beth K, Abishag N, Ben O. Computational Models for (M, K)-Quasi-*-Parahyponormal Operators. Pure Appl Math J. 2025;14(5):120-129. doi: 10.11648/j.pamj.20251405.13

Copy | Download

@article{10.11648/j.pamj.20251405.13,
  author = {Kiratu Beth and Ngoci Abishag and Obiero Ben},
  title = {Computational Models for (M, K)-Quasi-*-Parahyponormal Operators
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {14},
  number = {5},
  pages = {120-129},
  doi = {10.11648/j.pamj.20251405.13},
  url = {https://doi.org/10.11648/j.pamj.20251405.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20251405.13},
  abstract = {We study the two-parameter class of (M, k)-Quasi-∗-Parahyponormal operators on separable Hilbert spaces, which strictly enlarges the traditional parahyponormal and paranormal hierarchies. Analytically we prove three fundamental results: Every operator in the class has finite ascent and enjoys the single-valued extension property (SVEP); The Browder–Weyl partition holds, so Weyl’s theorem is valid; A non-trivial closed invariant subspace exists whenever the commutant contains a non-zero compact element. Complementing these theorems, we introduce a proposed computational framework that realises the abstract operators as large weighted-shift matrices, verifies the defining quadratic inequality, and computes eigenvalues as well aaccelerated pseudospectra. Together, the analytic results and the computational framework deepen the spectral theory of (M, k)-Quasi-∗-Parahyponormal Operators and supply the first large-scale numerical evidence for their structural properties.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Computational Models for (M, K)-Quasi-*-Parahyponormal Operators

AU  - Kiratu Beth
AU  - Ngoci Abishag
AU  - Obiero Ben
Y1  - 2025/09/25
PY  - 2025
N1  - https://doi.org/10.11648/j.pamj.20251405.13
DO  - 10.11648/j.pamj.20251405.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 120
EP  - 129
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20251405.13
AB  - We study the two-parameter class of (M, k)-Quasi-∗-Parahyponormal operators on separable Hilbert spaces, which strictly enlarges the traditional parahyponormal and paranormal hierarchies. Analytically we prove three fundamental results: Every operator in the class has finite ascent and enjoys the single-valued extension property (SVEP); The Browder–Weyl partition holds, so Weyl’s theorem is valid; A non-trivial closed invariant subspace exists whenever the commutant contains a non-zero compact element. Complementing these theorems, we introduce a proposed computational framework that realises the abstract operators as large weighted-shift matrices, verifies the defining quadratic inequality, and computes eigenvalues as well aaccelerated pseudospectra. Together, the analytic results and the computational framework deepen the spectral theory of (M, k)-Quasi-∗-Parahyponormal Operators and supply the first large-scale numerical evidence for their structural properties.

VL  - 14
IS  - 5
ER  -

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