Remarks on <i>A</i>-skew-adjoint, <i>A</i>-almost Similarity Equivalence and Other Operators in Hilbert Space

Isaiah Nalianya Sitati; Bernard Nzimbi; Stephen Luketero; Jairus Khalagai

doi:doi:10.11648/j.pamj.20170603.12

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Remarks on A-skew-adjoint, A-almost Similarity Equivalence and Other Operators in Hilbert Space

Isaiah Nalianya Sitati, Bernard Nzimbi, Stephen Luketero, Jairus Khalagai

Published in Pure and Applied Mathematics Journal (Volume 6, Issue 3)

Received: 28 April 2017 Accepted: 9 May 2017 Published: 29 June 2017

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Abstract

In this paper, notions of A-almost similarity and the Lie algebra of A-skew-adjoint operators in Hilbert space are introduced. In this context, A is a self-adjoint and an invertible operator. It is shown that A-almost similarity is an equivalence relation. Conditions under which A-almost similarity implies similarity are outlined and in which case their spectra is located. Conditions under which an A-skew adjoint operator reduces to a skew adjoint operator are also given. By relaxing some conditions on normal and unitary operators, new results on A -normal, binormal and A-binormal operators are proved. Finally A-skew adjoint operators are characterized and the relationship between A-self- adjoint and A-skew adjoint operators is given.

Published in	Pure and Applied Mathematics Journal (Volume 6, Issue 3)
DOI	10.11648/j.pamj.20170603.12
Page(s)	101-107
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

Skew-adjoint, A-skew-adjoint, A-almost Similarity, Hilbert Space, A-Normal and Binormal

References

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APA Style

Isaiah Nalianya Sitati, Bernard Nzimbi, Stephen Luketero, Jairus Khalagai. (2017). Remarks on A-skew-adjoint, A-almost Similarity Equivalence and Other Operators in Hilbert Space. Pure and Applied Mathematics Journal, 6(3), 101-107. https://doi.org/10.11648/j.pamj.20170603.12

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ACS Style

Isaiah Nalianya Sitati; Bernard Nzimbi; Stephen Luketero; Jairus Khalagai. Remarks on A-skew-adjoint, A-almost Similarity Equivalence and Other Operators in Hilbert Space. Pure Appl. Math. J. 2017, 6(3), 101-107. doi: 10.11648/j.pamj.20170603.12

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AMA Style

Isaiah Nalianya Sitati, Bernard Nzimbi, Stephen Luketero, Jairus Khalagai. Remarks on A-skew-adjoint, A-almost Similarity Equivalence and Other Operators in Hilbert Space. Pure Appl Math J. 2017;6(3):101-107. doi: 10.11648/j.pamj.20170603.12

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@article{10.11648/j.pamj.20170603.12,
  author = {Isaiah Nalianya Sitati and Bernard Nzimbi and Stephen Luketero and Jairus Khalagai},
  title = {Remarks on A-skew-adjoint, A-almost Similarity Equivalence and Other Operators in Hilbert Space},
  journal = {Pure and Applied Mathematics Journal},
  volume = {6},
  number = {3},
  pages = {101-107},
  doi = {10.11648/j.pamj.20170603.12},
  url = {https://doi.org/10.11648/j.pamj.20170603.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20170603.12},
  abstract = {In this paper, notions of A-almost similarity and the Lie algebra of A-skew-adjoint operators in Hilbert space are introduced. In this context, A is a self-adjoint and an invertible operator. It is shown that A-almost similarity is an equivalence relation. Conditions under which A-almost similarity implies similarity are outlined and in which case their spectra is located. Conditions under which an A-skew adjoint operator reduces to a skew adjoint operator are also given. By relaxing some conditions on normal and unitary operators, new results on A -normal, binormal and A-binormal operators are proved. Finally A-skew adjoint operators are characterized and the relationship between A-self- adjoint and A-skew adjoint operators is given.},
 year = {2017}
}

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TY  - JOUR
T1  - Remarks on A-skew-adjoint, A-almost Similarity Equivalence and Other Operators in Hilbert Space
AU  - Isaiah Nalianya Sitati
AU  - Bernard Nzimbi
AU  - Stephen Luketero
AU  - Jairus Khalagai
Y1  - 2017/06/29
PY  - 2017
N1  - https://doi.org/10.11648/j.pamj.20170603.12
DO  - 10.11648/j.pamj.20170603.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 101
EP  - 107
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20170603.12
AB  - In this paper, notions of A-almost similarity and the Lie algebra of A-skew-adjoint operators in Hilbert space are introduced. In this context, A is a self-adjoint and an invertible operator. It is shown that A-almost similarity is an equivalence relation. Conditions under which A-almost similarity implies similarity are outlined and in which case their spectra is located. Conditions under which an A-skew adjoint operator reduces to a skew adjoint operator are also given. By relaxing some conditions on normal and unitary operators, new results on A -normal, binormal and A-binormal operators are proved. Finally A-skew adjoint operators are characterized and the relationship between A-self- adjoint and A-skew adjoint operators is given.
VL  - 6
IS  - 3
ER  -

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Author Information

Isaiah Nalianya Sitati

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

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

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

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

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APA Style

Isaiah Nalianya Sitati, Bernard Nzimbi, Stephen Luketero, Jairus Khalagai. (2017). Remarks on A-skew-adjoint, A-almost Similarity Equivalence and Other Operators in Hilbert Space. Pure and Applied Mathematics Journal, 6(3), 101-107. https://doi.org/10.11648/j.pamj.20170603.12

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ACS Style

Isaiah Nalianya Sitati; Bernard Nzimbi; Stephen Luketero; Jairus Khalagai. Remarks on A-skew-adjoint, A-almost Similarity Equivalence and Other Operators in Hilbert Space. Pure Appl. Math. J. 2017, 6(3), 101-107. doi: 10.11648/j.pamj.20170603.12

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AMA Style

Isaiah Nalianya Sitati, Bernard Nzimbi, Stephen Luketero, Jairus Khalagai. Remarks on A-skew-adjoint, A-almost Similarity Equivalence and Other Operators in Hilbert Space. Pure Appl Math J. 2017;6(3):101-107. doi: 10.11648/j.pamj.20170603.12

Copy | Download

@article{10.11648/j.pamj.20170603.12,
  author = {Isaiah Nalianya Sitati and Bernard Nzimbi and Stephen Luketero and Jairus Khalagai},
  title = {Remarks on A-skew-adjoint, A-almost Similarity Equivalence and Other Operators in Hilbert Space},
  journal = {Pure and Applied Mathematics Journal},
  volume = {6},
  number = {3},
  pages = {101-107},
  doi = {10.11648/j.pamj.20170603.12},
  url = {https://doi.org/10.11648/j.pamj.20170603.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20170603.12},
  abstract = {In this paper, notions of A-almost similarity and the Lie algebra of A-skew-adjoint operators in Hilbert space are introduced. In this context, A is a self-adjoint and an invertible operator. It is shown that A-almost similarity is an equivalence relation. Conditions under which A-almost similarity implies similarity are outlined and in which case their spectra is located. Conditions under which an A-skew adjoint operator reduces to a skew adjoint operator are also given. By relaxing some conditions on normal and unitary operators, new results on A -normal, binormal and A-binormal operators are proved. Finally A-skew adjoint operators are characterized and the relationship between A-self- adjoint and A-skew adjoint operators is given.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Remarks on A-skew-adjoint, A-almost Similarity Equivalence and Other Operators in Hilbert Space
AU  - Isaiah Nalianya Sitati
AU  - Bernard Nzimbi
AU  - Stephen Luketero
AU  - Jairus Khalagai
Y1  - 2017/06/29
PY  - 2017
N1  - https://doi.org/10.11648/j.pamj.20170603.12
DO  - 10.11648/j.pamj.20170603.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 101
EP  - 107
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20170603.12
AB  - In this paper, notions of A-almost similarity and the Lie algebra of A-skew-adjoint operators in Hilbert space are introduced. In this context, A is a self-adjoint and an invertible operator. It is shown that A-almost similarity is an equivalence relation. Conditions under which A-almost similarity implies similarity are outlined and in which case their spectra is located. Conditions under which an A-skew adjoint operator reduces to a skew adjoint operator are also given. By relaxing some conditions on normal and unitary operators, new results on A -normal, binormal and A-binormal operators are proved. Finally A-skew adjoint operators are characterized and the relationship between A-self- adjoint and A-skew adjoint operators is given.
VL  - 6
IS  - 3
ER  -

Copy | Download