On Some <i>n</i>-Involution and <i>k</i>-Potent Operators on Hilbert Spaces

Bernard Mutuku Nzimbi; Beth Nyambura Kiratu; Stephen Wanyonyi Luketero

doi:doi:10.11648/j.mcs.20170206.11

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On Some n-Involution and k-Potent Operators on Hilbert Spaces

Bernard Mutuku Nzimbi, Beth Nyambura Kiratu, Stephen Wanyonyi Luketero

Published in Mathematics and Computer Science (Volume 2, Issue 6)

Received: 15 May 2017 Accepted: 3 June 2017 Published: 14 November 2017

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Abstract

In this paper, we survey various results concerning -involution operators and -potent operators in Hilbert spaces. We gain insight by studying the operator equation , with where . We study the structure of such operators and attempt to gain information about the structure of closely related operators, associated operators and the attendant spectral geometry. The paper concludes with some applications in integral equations.

Published in	Mathematics and Computer Science (Volume 2, Issue 6)
DOI	10.11648/j.mcs.20170206.11
Page(s)	79-88
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), 2017. Published by Science Publishing Group

Keywords

n-Involution, Idempotent, Spectral Radius, Twist, Invection, Q-Equivalence

References

[1]	S. Furtado, Products of two involutions with prescribed eigenvalues and some applications, Linear Algebra and its Applications, 429(2008), 1663-1678.
[2]	T. Furuta, Invitation to linear operators: from matrices to bounded linear operators on a Hilbert space, Taylor Francis, London, 2001.
[3]	M. I. Kadets and K. E. Kaibkhanov, Continuation of a linear operator to an involution, Mathematical Notes 61, No. 5 (1997), 560–565.
[4]	Nikolai Karapetiants and Stefan Samko, Equations with involutive operators, Springer Science + Business Media, LLC, New York, 2001.
[5]	C. S. Kubrusly, An introduction to models and decompositions in operator theory, Birkh¨auser, Boston, 1997.
[6]	Mao-Lin Liang and Li-Fang Dai, The solvability conditions of matrix equations with involutions, Electronic Journal of Linear Algebra, Vol. 22 (2011), 1138-1147.
[7]	B. M. Nzimbi, G. P. Pokhariyal, and S. K. Moindi, A note on metric equivalence of some operators, Far East J. of Math. Sci. (FJMS) 75, No. 2 (2013), 301–318.
[8]	S. K. Singh, G. Mukherjee, and M. Kumar Roy, -range, -range of operators in a Hilbert space, International J. of Engineering and Sciences (IJES) 3, Issue 8 (2014), 26–35.
[9]	T. Yongge, A disjoint idempotent decomposition for linear combinations produced from two commutative tripotent matrices and its applications, Linear and Multilinear Algebra, Vol. 59, No. 11(2011), 1237-1246.
[10]	C. Xu, On the idempotency, involution and nilpotency of a linear combination of two matrices, Linear and Multilinear Algebra, Vol. 63, No. 8(2015), 1664-1677.

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APA Style

Bernard Mutuku Nzimbi, Beth Nyambura Kiratu, Stephen Wanyonyi Luketero. (2017). On Some n-Involution and k-Potent Operators on Hilbert Spaces. Mathematics and Computer Science, 2(6), 79-88. https://doi.org/10.11648/j.mcs.20170206.11

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ACS Style

Bernard Mutuku Nzimbi; Beth Nyambura Kiratu; Stephen Wanyonyi Luketero. On Some n-Involution and k-Potent Operators on Hilbert Spaces. Math. Comput. Sci. 2017, 2(6), 79-88. doi: 10.11648/j.mcs.20170206.11

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AMA Style

Bernard Mutuku Nzimbi, Beth Nyambura Kiratu, Stephen Wanyonyi Luketero. On Some n-Involution and k-Potent Operators on Hilbert Spaces. Math Comput Sci. 2017;2(6):79-88. doi: 10.11648/j.mcs.20170206.11

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@article{10.11648/j.mcs.20170206.11,
  author = {Bernard Mutuku Nzimbi and Beth Nyambura Kiratu and Stephen Wanyonyi Luketero},
  title = {On Some n-Involution and k-Potent Operators on Hilbert Spaces},
  journal = {Mathematics and Computer Science},
  volume = {2},
  number = {6},
  pages = {79-88},
  doi = {10.11648/j.mcs.20170206.11},
  url = {https://doi.org/10.11648/j.mcs.20170206.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20170206.11},
  abstract = {In this paper, we survey various results concerning -involution operators and -potent operators in Hilbert spaces. We gain insight by studying the operator equation , with  where . We study the structure of such operators and attempt to gain information about the structure of closely related operators, associated operators and the attendant spectral geometry. The paper concludes with some applications in integral equations.},
 year = {2017}
}

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TY  - JOUR
T1  - On Some n-Involution and k-Potent Operators on Hilbert Spaces
AU  - Bernard Mutuku Nzimbi
AU  - Beth Nyambura Kiratu
AU  - Stephen Wanyonyi Luketero
Y1  - 2017/11/14
PY  - 2017
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PB  - Science Publishing Group
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AB  - In this paper, we survey various results concerning -involution operators and -potent operators in Hilbert spaces. We gain insight by studying the operator equation , with  where . We study the structure of such operators and attempt to gain information about the structure of closely related operators, associated operators and the attendant spectral geometry. The paper concludes with some applications in integral equations.
VL  - 2
IS  - 6
ER  -

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Author Information

Bernard Mutuku Nzimbi

School of Mathematics, College of Biological and Physical Sciences, University of Nairobi, Nairobi, Kenya
Beth Nyambura Kiratu

Department of Pure and Applied Mathematics, Faculty of Applied Sciences and Technology, Technical University of Kenya, Nairobi, Kenya
Stephen Wanyonyi Luketero

School of Mathematics, College of Biological and Physical Sciences, University of Nairobi, Nairobi, Kenya

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APA Style

Bernard Mutuku Nzimbi, Beth Nyambura Kiratu, Stephen Wanyonyi Luketero. (2017). On Some n-Involution and k-Potent Operators on Hilbert Spaces. Mathematics and Computer Science, 2(6), 79-88. https://doi.org/10.11648/j.mcs.20170206.11

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ACS Style

Bernard Mutuku Nzimbi; Beth Nyambura Kiratu; Stephen Wanyonyi Luketero. On Some n-Involution and k-Potent Operators on Hilbert Spaces. Math. Comput. Sci. 2017, 2(6), 79-88. doi: 10.11648/j.mcs.20170206.11

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AMA Style

Bernard Mutuku Nzimbi, Beth Nyambura Kiratu, Stephen Wanyonyi Luketero. On Some n-Involution and k-Potent Operators on Hilbert Spaces. Math Comput Sci. 2017;2(6):79-88. doi: 10.11648/j.mcs.20170206.11

Copy | Download

@article{10.11648/j.mcs.20170206.11,
  author = {Bernard Mutuku Nzimbi and Beth Nyambura Kiratu and Stephen Wanyonyi Luketero},
  title = {On Some n-Involution and k-Potent Operators on Hilbert Spaces},
  journal = {Mathematics and Computer Science},
  volume = {2},
  number = {6},
  pages = {79-88},
  doi = {10.11648/j.mcs.20170206.11},
  url = {https://doi.org/10.11648/j.mcs.20170206.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20170206.11},
  abstract = {In this paper, we survey various results concerning -involution operators and -potent operators in Hilbert spaces. We gain insight by studying the operator equation , with  where . We study the structure of such operators and attempt to gain information about the structure of closely related operators, associated operators and the attendant spectral geometry. The paper concludes with some applications in integral equations.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - On Some n-Involution and k-Potent Operators on Hilbert Spaces
AU  - Bernard Mutuku Nzimbi
AU  - Beth Nyambura Kiratu
AU  - Stephen Wanyonyi Luketero
Y1  - 2017/11/14
PY  - 2017
N1  - https://doi.org/10.11648/j.mcs.20170206.11
DO  - 10.11648/j.mcs.20170206.11
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 79
EP  - 88
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20170206.11
AB  - In this paper, we survey various results concerning -involution operators and -potent operators in Hilbert spaces. We gain insight by studying the operator equation , with  where . We study the structure of such operators and attempt to gain information about the structure of closely related operators, associated operators and the attendant spectral geometry. The paper concludes with some applications in integral equations.
VL  - 2
IS  - 6
ER  -

Copy | Download