Pure and Applied Mathematics Journal

| Peer-Reviewed |

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

Received: 28 April 2017    Accepted: 09 May 2017    Published: 29 June 2017
Views:       Downloads:

Share This Article

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.

DOI 10.11648/j.pamj.20170603.12
Published in Pure and Applied Mathematics Journal (Volume 6, Issue 3, June 2017)
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), 2024. Published by Science Publishing Group

Keywords

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

References
[1] Cassier G, Mahzouli H. and Zeroiali E. H, Generalized Toeplitz operators and cyclic operators, Oper. Theor. Advances and Applications 153 (2004): 03-122.
[2] Isaiah N. Sitati, Bernard M. Nzimbi, Stephen W. Luketero, Jairus M. Khalagai. On A-Self- Adjoint, A-Unitary Operators and Quasiaffinities. Mathematics and Computer Science. Vol. 1, No. 3, 2016, pp. 36-41. doi: 10.11648/j.mcs.20160103.11.
[3] Jibril A. A, On almost similar operators, Arabian J. Sci. Engrg. 21 (1996), pp. 443-449.
[4] Kubrusly C. S., An Introduction to Models and Decompositions in Operator Theory. Birkha ̈users, Boston, (1997).
[5] Kubrusly C. S., Hilbert Space Operators, Birkhausers, Basel, Boston, (2003).
[6] Lins B, Meade P, Mehl C and Rodman L. Normal Matrices and Polar decompositions in infinite Inner Products. Linear and Multilinear algebra, 49: 45-89, (2001).
[7] Mehl C. and Rodman L. Classes of Normal Matrices in infinite Inner Products. Linear algebra Appl, 336: 71-98, (2001).
[8] Mostafazadeh A., Pseudo-Hermiticity versus PT-symmetry, III, Equivalance of pseudo-Hermiticity and the presence of antilinear symmetries, J. Math. Phys. 43 (8) (2002), 3944-3951.
[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.
[10] Nzimbi B. M, Pokhariyal G. P and Moindi S. K, A note on A-self-adjoint and A-Skew adjoint Operators, Pioneer Journal of Mathematics and Mathematical sciences, (2013), 1-36.
[11] Nzimbi B. M, Pokhariyal G. P and Moindi S. K, A note on Metric Equivalence of Some Operators, Far East Journal of Mathematical sciences, Vol 75, No. 2 (2013), 301-318.
[12] Nzimbi B. M., Khalagai J. M. and Pokhariyal G. P., A note on similarity, almost similarity and equivalence of operators, Far East J. Math. Sci. (FMJS) 28 (2) (2008), 305-317.
[13] Nzimbi B. M, Luketero S. W, Sitati I. N, Musundi S. W and Mwenda E, On Almost Similarity and Metric Equivalence of Operators, Accepted to be published by Pioneer Journal of Mathematics and Mathematical sciences (June 14, 2016).
[14] Patel S. M., A note on quasi-isometries II, Glasnik Matematicki 38 (58) (2003), 111-120.
[15] Rehder Wulf, On the product of self-adjoint operators, Internat. J. Math. and Math. Sci 5 (4) (1982), 813-816.
[16] Rudin W, Functional Analysis, 2nd ed., International Series in Pure and Applied Math., Mc Graw-Hill’s, Boston, (1991).
[17] Suciu L, Some invariant subspaces of A-contractions and applications, Extracta Mathematicae 21 (3) (2006), 221-247.
[18] Sz-Nagy B, Foias C, Bercovivi H and Kerchy L, Harmonic Analysis of Operators on Hilbert Space, Springer New York Dordrecht London (2010).
[19] Tucanak M and Weiss G, Observation and Control for Operator Semigroups, Birkhauser, Verlag, Basel, (2009).
[20] Virtanen J. A: Operator Theory Fall (2007).
[21] Yeung Y. H, Li C. K and L. Rodman, on H-unitary and Block Toeplitz H-normal operators, H-unitary and Lorentz matrices: A review, Preprint.
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

Cite This Article
  • 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

    Copy | Download

    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

    Copy | Download

    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://download.sciencepg.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

  • Sections