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

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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[9] Musundi S. W, Sitati I. N, Nzimbi B. M, Murwayi A. L, On Almost Similarity Operator Equivalence Relation, International Journal of Research and Reviews in Applied Sciences, Vol 15, No. 3 (2012), pp. 293-299.
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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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    T1  - Remarks on A-skew-adjoint, A-almost Similarity Equivalence and Other Operators in Hilbert Space
    AU  - Isaiah Nalianya Sitati
    AU  - Bernard Nzimbi
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    JF  - Pure and Applied Mathematics Journal
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    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
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Author Information
  • School of Mathematics, College of Biological and Physical Sciences, University of Nairobi, Nairobi, Kenya

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

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

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

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