On (α, β)–Almost Similar Operators in Hilbert Spaces

Beatrice Obiero Adhiambo; Victor Wanjala

doi:doi:10.11648/j.pamj.20261502.13

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On (α, β)–Almost Similar Operators in Hilbert Spaces

Beatrice Obiero Adhiambo

, Victor Wanjala^*

Published in Pure and Applied Mathematics Journal (Volume 15, Issue 2)

Received: 16 March 2026 Accepted: 25 March 2026 Published: 24 April 2026

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Abstract

This paper introduces and investigates a novel generalization of operator similarity, termed (α,β)--almost similarity, which extends the concept of almost similar operators by incorporating two real parameters. We establish fundamental properties of this new equivalence relation, demonstrating that it forms an equivalence class on the space of bounded linear operators on a Hilbert space. Key results include the invariance of spectrum, point spectrum, and approximate point spectrum under this relation. The study also defines the class of (α,β)-𝔗 operators, an expansion of the classical 𝔗-operator concept, and explores its relationship with (α,β)--almost similarity. Furthermore, we analyze the connections between similarity, unitary equivalence, and (α,β)--almost similarity, providing conditions under which these relations coincide, particularly for self-adjoint and projection operators. The results contribute to the broader understanding of operator equivalence relations and their spectral implications.

Published in	Pure and Applied Mathematics Journal (Volume 15, Issue 2)
DOI	10.11648/j.pamj.20261502.13
Page(s)	29-34
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), 2026. Published by Science Publishing Group

Keywords

α–almost Similar, Almost Similar, Unitarily Equivalent, Self-adjoint Operator

References

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[2]	Dehimi, S., Operators similar to their adjoints, D. Sc. Thesis, University of Oran1 Ahmed Ben Bella, (2017).
[3]	Isaiah, N. S., Sammi, W. M., B. M. Nzimbi and KiketeW. D., A note on quasi-similarity in Hilbert spaces, International Journal of Math. Archive-6(7) (2015), 49-55.
[4]	Berberian, S. K., Introduction to Hilbert space, Oxforduniversity Press, New York, (1961).
[5]	I. N. Sitati, B. M. Nzimbi, Stephen L. and JairusK., Remarks on A-skew-adjoint, A-almost similarity equivalence and other operators in Hilbert space, Pure and Applied Mathematics Journal, 6(3) (2017), 101-107. https://doi.org/10.11648/j.pamj.20170603.12
[6]	Kipkemoi, T. S., On almost similarity and other related equivalence relations of operators in Hilbert spaces, M. Sc. Thesis, University of Nairobi, (2016).
[7]	Jibril, A. A., On almost similar operators, Arabian J. Sci. Engrg., 21 (1996), 434-449.
[8]	Campbell, S. L. and Gellar, R., Linear operators for which T∗T and TT∗ commute, Trans. Of the Amer. Math. Soc., 226 (1977), 305-319. https://doi.org/10.2307/1997907
[9]	Kreyszig, E., Introductory functional analysis with application, Wiley, New York, (1978).
[10]	Musundi, S.W., Sitati, N. I., Nzimbi, B. M and Murwayi, A. L., On almost similarity operator equivalence relation, IJERRAS 15(3) (2013), 293-299.
[11]	Conway, J. B., A Course in Functional Analysis, 2nd ed., Springer-Verlag, New York, (1990). https://doi.org/10.1007/978-1-4757-3828-5
[12]	Davidson, K. R., C∗-Algebras by Example, Fields Institute Monographs, Vol. 6, American Mathematical Society, Providence, RI, (1996). https://doi.org/10.1090/fim/006
[13]	Fillmore, P. A., On similarity and the diagonal of a matrix, Linear Algebra and its Applications, 6 (1973), 177-186. https://doi.org/10.1016/0024-3795(73)90017-3
[14]	Halmos, P. R., A Hilbert Space Problem Book, 2nd ed., Springer-Verlag, New York, (1982). https://doi.org/10.1007/978-1-4684-9330-6
[15]	Simon, B., Orthogonal Polynomials on the Unit Circle: Part 1: Classical Theory, American Mathematical Society Colloquium Publications, Vol. 54, Providence, RI, (2005). https://doi.org/10.1090/coll/054.1
[16]	Weidmann, J., Linear Operators in Hilbert Spaces, Springer-Verlag, New York, (1980). https://doi.org/10.1007/978-1-4612-6027-1

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APA Style

Adhiambo, B. O., Wanjala, V. (2026). On (α, β)–Almost Similar Operators in Hilbert Spaces. Pure and Applied Mathematics Journal, 15(2), 29-34. https://doi.org/10.11648/j.pamj.20261502.13

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ACS Style

Adhiambo, B. O.; Wanjala, V. On (α, β)–Almost Similar Operators in Hilbert Spaces. Pure Appl. Math. J. 2026, 15(2), 29-34. doi: 10.11648/j.pamj.20261502.13

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AMA Style

Adhiambo BO, Wanjala V. On (α, β)–Almost Similar Operators in Hilbert Spaces. Pure Appl Math J. 2026;15(2):29-34. doi: 10.11648/j.pamj.20261502.13

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@article{10.11648/j.pamj.20261502.13,
  author = {Beatrice Obiero Adhiambo and Victor Wanjala},
  title = {On (α, β)–Almost Similar Operators in Hilbert Spaces},
  journal = {Pure and Applied Mathematics Journal},
  volume = {15},
  number = {2},
  pages = {29-34},
  doi = {10.11648/j.pamj.20261502.13},
  url = {https://doi.org/10.11648/j.pamj.20261502.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20261502.13},
  abstract = {This paper introduces and investigates a novel generalization of operator similarity, termed (α,β)--almost similarity, which extends the concept of almost similar operators by incorporating two real parameters. We establish fundamental properties of this new equivalence relation, demonstrating that it forms an equivalence class on the space of bounded linear operators on a Hilbert space. Key results include the invariance of spectrum, point spectrum, and approximate point spectrum under this relation. The study also defines the class of (α,β)-𝔗 operators, an expansion of the classical 𝔗-operator concept, and explores its relationship with (α,β)--almost similarity. Furthermore, we analyze the connections between similarity, unitary equivalence, and (α,β)--almost similarity, providing conditions under which these relations coincide, particularly for self-adjoint and projection operators. The results contribute to the broader understanding of operator equivalence relations and their spectral implications.},
 year = {2026}
}

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TY  - JOUR
T1  - On (α, β)–Almost Similar Operators in Hilbert Spaces
AU  - Beatrice Obiero Adhiambo
AU  - Victor Wanjala
Y1  - 2026/04/24
PY  - 2026
N1  - https://doi.org/10.11648/j.pamj.20261502.13
DO  - 10.11648/j.pamj.20261502.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 29
EP  - 34
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20261502.13
AB  - This paper introduces and investigates a novel generalization of operator similarity, termed (α,β)--almost similarity, which extends the concept of almost similar operators by incorporating two real parameters. We establish fundamental properties of this new equivalence relation, demonstrating that it forms an equivalence class on the space of bounded linear operators on a Hilbert space. Key results include the invariance of spectrum, point spectrum, and approximate point spectrum under this relation. The study also defines the class of (α,β)-𝔗 operators, an expansion of the classical 𝔗-operator concept, and explores its relationship with (α,β)--almost similarity. Furthermore, we analyze the connections between similarity, unitary equivalence, and (α,β)--almost similarity, providing conditions under which these relations coincide, particularly for self-adjoint and projection operators. The results contribute to the broader understanding of operator equivalence relations and their spectral implications.
VL  - 15
IS  - 2
ER  -

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Author Information

Beatrice Obiero Adhiambo

Department of Mathematics and Computer, Rongo University, Rongo, Kenya

http://orcid.org/0000-0002-0923-3447
Victor Wanjala

Department of Mathematics and Physical Sciences, Maasai Mara University, Narok, Kenya

Contact Email

http://orcid.org/0000-0002-9207-5573

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APA Style

Adhiambo, B. O., Wanjala, V. (2026). On (α, β)–Almost Similar Operators in Hilbert Spaces. Pure and Applied Mathematics Journal, 15(2), 29-34. https://doi.org/10.11648/j.pamj.20261502.13

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ACS Style

Adhiambo, B. O.; Wanjala, V. On (α, β)–Almost Similar Operators in Hilbert Spaces. Pure Appl. Math. J. 2026, 15(2), 29-34. doi: 10.11648/j.pamj.20261502.13

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AMA Style

Adhiambo BO, Wanjala V. On (α, β)–Almost Similar Operators in Hilbert Spaces. Pure Appl Math J. 2026;15(2):29-34. doi: 10.11648/j.pamj.20261502.13

Copy | Download

@article{10.11648/j.pamj.20261502.13,
  author = {Beatrice Obiero Adhiambo and Victor Wanjala},
  title = {On (α, β)–Almost Similar Operators in Hilbert Spaces},
  journal = {Pure and Applied Mathematics Journal},
  volume = {15},
  number = {2},
  pages = {29-34},
  doi = {10.11648/j.pamj.20261502.13},
  url = {https://doi.org/10.11648/j.pamj.20261502.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20261502.13},
  abstract = {This paper introduces and investigates a novel generalization of operator similarity, termed (α,β)--almost similarity, which extends the concept of almost similar operators by incorporating two real parameters. We establish fundamental properties of this new equivalence relation, demonstrating that it forms an equivalence class on the space of bounded linear operators on a Hilbert space. Key results include the invariance of spectrum, point spectrum, and approximate point spectrum under this relation. The study also defines the class of (α,β)-𝔗 operators, an expansion of the classical 𝔗-operator concept, and explores its relationship with (α,β)--almost similarity. Furthermore, we analyze the connections between similarity, unitary equivalence, and (α,β)--almost similarity, providing conditions under which these relations coincide, particularly for self-adjoint and projection operators. The results contribute to the broader understanding of operator equivalence relations and their spectral implications.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - On (α, β)–Almost Similar Operators in Hilbert Spaces
AU  - Beatrice Obiero Adhiambo
AU  - Victor Wanjala
Y1  - 2026/04/24
PY  - 2026
N1  - https://doi.org/10.11648/j.pamj.20261502.13
DO  - 10.11648/j.pamj.20261502.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 29
EP  - 34
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20261502.13
AB  - This paper introduces and investigates a novel generalization of operator similarity, termed (α,β)--almost similarity, which extends the concept of almost similar operators by incorporating two real parameters. We establish fundamental properties of this new equivalence relation, demonstrating that it forms an equivalence class on the space of bounded linear operators on a Hilbert space. Key results include the invariance of spectrum, point spectrum, and approximate point spectrum under this relation. The study also defines the class of (α,β)-𝔗 operators, an expansion of the classical 𝔗-operator concept, and explores its relationship with (α,β)--almost similarity. Furthermore, we analyze the connections between similarity, unitary equivalence, and (α,β)--almost similarity, providing conditions under which these relations coincide, particularly for self-adjoint and projection operators. The results contribute to the broader understanding of operator equivalence relations and their spectral implications.
VL  - 15
IS  - 2
ER  -

Copy | Download