International Journal of Modern Science and Technology · ISSN 2456-0235

Research Article · Mathematical Sciences · Volume 3, Issue 10 · Oct 2018 · Pages 203–207

Certain Aspects of Normal Classes of Hilbert Space Operators

N. B. Okelo

  • N. B. Okelo: School of Mathematics and Actuarial Science, Jaramogi Oginga Odinga University of Science and Technology, P. O. Box 210-40601, Bondo-Kenya.

Abstract

Let T be a Quasi - * - class A normal operator on a complex Hilbert space H. In this paper, we prove that if E is the Riesz idempotent for a non-zero isolated point λ of the spectrum of T ∈ B(H) of Quasi * - class A normal operator, then E is self-adjoint and EH = ker(T − λ) = ker (T − λ)*. We will also prove a necessary and sufficient condition for T ⊗ S to be quasi - * - class A normal where T and S are both non-zero operators.

Keywords

Paranormal operators; Weyl’s theorem; * - class A normal operators; Quasi - * - class A

References

  1. Sun LL, Gao W, Zhang MM, Li C, Wang AG, Su YL, Ji TF. Composition and Antioxidant Activity of the Anthocyanins of the Fruit of Berberis heteropoda Schrenk. Molecules 2014;19:19078-96.
  2. Aluthge A, On p - hyponormal operators for 0 < p < 1. Integral Equations Operator Theory 1990;13(3):307-15.
  3. Aluthge A, Some generalized theorems on p - hyponormal operators . Integral Equations Operator Theory 1996;24(4):497-501.
  4. Arora SC, and Arora P. On p - quasihyponormal operators for 0 < p < 1. Yokohama Math J 1993;41:25-9.
  5. Ando T. Operators with a norm condition , Acta Sci Math (Szeged) 1973; 33:169- 78.
  6. Ando T. On some operator ineq ualities. Math Ann 1987;279(1):157-59.
  7. Campbell SL, Gupta B C. On k - quasihyponormal operators . Math. Joponica 1978;23:185-89.
  8. Rhaly HC. Posinormal Operators . J Math Soc Japan 1974;46:587-605.
  9. Duggal BP, Jeon IH, Kim IH. Weyl’s theorem in the class of alg ebraically p - hyponormal operators . Comment Math Prace Mat 2000;40:49-56.
  10. Fututa T, Ito M, Yamazaki T. A subclass of paranormal including class of log - hyponormal and several related classes . Scientiae Mathematicae 1998;1:389-403.
  11. Hou JC. On the tensor products of operators. Acta M ath Sinica 1992;9(2):195-202.
  12. Jeon IH, Kim IH. On operators satisfying T*|T2|T ~ T*|T|2T , Linear Algebra Appl 2006;418(2):854-62.
  13. Shen JL, Zuo F, Yang CS. On Operators Satisfying T*|T2|T ~ T*|T*|2T . Acta Mathematica Sinica 2010;26:2109-16.
  14. Mecheri S. Isolated points of spectrum of k - quasi - * - class A operators. Studia Math 2012;208(1):87-96.
  15. Shen JL, Zuo F, Yang CS. On operators satisfying T*|T2|T ~ T*|T*2|T Part II. Acta Math. Sinica. 2013;32:3100-40.
  16. Stampfli JG. Hyponormal operators and spectral density . Trans Amer Math Soc 1965;117:469-76.
  17. Saito T. Hyponormal operators and related topics, Lectures on operator algebras Lecture Notes in Math., Springer, Berlin; 1972.
  18. Stochel J. Seminormality of operators from their tensor product. Proc Amer Math Soc 1996;124(1):135-40.
  19. Uchiyama A. On isolated points of the spectrum of paranomal operators . Integral Equations Operator Theory 2006;55:145 - 51.
  20. Xia D. Spectral Theory of Hyponormal Operators. Birkhauser Verlag, Boston; 1983. ©2018 The Authors. Published by G. J. Publications under the CC BY license. 207.
  21. Ringrose J R. Compact Non -self-adjoint operators, Van Nostrand Reinhold, London; 2015.
  22. Schatten R. Norm ideals of completely continuous operators, Springer-Verlag, Berlin; 2017.
  23. Taylor AE, Lay DC. Introduction to functional analysis, 2nd ed., John Wiley and Sons, New York; 2014.
  24. Okelo NB; Agure JO, Ambogo DO. Norms of elementary operators and characterization of Norm -Attainable operators. Int J Math Anal 2010;4:1197- 204.
  25. Vijayabalaji S, Shyamsundar G. Interval- valued intuitionistic fuzzy transition matrices, Int J Mod Sci Technol 2016; 1(2):47-51.
  26. Judith J O, Okelo NB, Roy K, Onyango T. Numerical Solutions of Mathematical Model on Effects of Biological Control on Cereal Aphid Population Dynamics, IJMST, 2016; 1(4): 138-143.
  27. Judith J O, Okelo NB, Roy K, Onyango T. Construction and Qualitative Analysis of Mathematical Model for Biological Control on C ereal Aphid Population Dynamics. Int J Mod Sci Technol 2016;1(5):150-58.
  28. Vijayabalaji S, Sathiyaseelan N. Interval- Valued Product Fuzzy Soft Matrices and its Application in Decision Making. Int J Mod Sci Technol 2016;1(7):159-63.
  29. Chinnadurai V, Bharathivelan K. Cubic Ideals in Near Subtraction Semigroups. Int J Mod Sci Technol 2016;1(8):276-82.
  30. Okello B, Okelo NB, Ongati O. Characterization of Norm Inequa lities for Elementary Operators . Int J Mod Sci Technol 2017;2(3):81-4.
  31. Wafula AM, Okelo NB, Ongati O. Norms of Normally R epresented Elementary Operators. Int J Mod Sci Technol 2018;3(1):10-6.