ISSN 2456-0235

International Journal of Modern Science and Technology

INDEXED IN 

​​​​​​​​​​​​​October 2018, Vol. 3, No. 10, pp. 203-207. 

​​​Certain Aspects of Normal Classes of Hilbert Space Operator

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

​​*Corresponding author’s e-mail: bnyaare@yahoo.com

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 normal operators.

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 inequalities. Math Ann 1987;279(1):157-59.
  7. Campbell SL, Gupta BC. 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 algebraically 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 Math 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.
  21. Ringrose JR. 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 Cereal 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 Inequalities for Elementary Operators. Int J Mod Sci Technol 2017;2(3):81-4.
  31. Wafula AM, Okelo NB, Ongati O. Norms of Normally Represented Elementary Operators. Int J Mod Sci Technol 2018;3(1):10-6.