​​​​​​​​​February 2018, Vol. 3, No 2, pp 27-32. 

​​Necessary and Sufficient conditions for Normality of Operators in Hilbert spaces

A. M. Wafula, N. B. Okelo*, O. Ongati
School of Mathematics and Actuarial Science, Jaramogi Oginga Odinga University of Science and Technology, P.O. Box 2010-4060, Bondo-Kenya.
​​​*Corresponding author’s e-mail: bnyaare@yahoo.com

​Abstract

​Characterization of normality is an interesting aspect for Hilbert space operators. In this paper, we have shown that for an operator A to be normal, it is necessary that A = A*. It is also sufficient that for an operator A to be normal then the condition AA* = A*A holds. Moreover, for an inner derivation, we conjecture that the property ᵟA=ᵟA* is necessary for its normality.

Keywords: Adjoint Operator; Normal operators; Posinormal operators; Positive operators.

​References

  1. Barraa M, Boumazgour M. A Lower bound of the norm of the operator X → AXB+BXA. Extracta Math. 2001;16:223-227.
  2. Blanco A, Boumazgour M, Ransford T. On the Norm of elementary operators. J London Math Soc. 2004;70:479-498.
  3. Cabrera M, Rodriguez A.  Nondegenerately ultraprint Jordan Banach algebras. Proc London Math Soc. 1994;69:576-604.
  4. Einsiedler M, Ward T. Functional Analysis notes. Lecture notes series; 2012.
  5. Landsman NP. C*-Algebras and Quantum mechanics. Lecture notes; 1998.
  6. Mathieu M. Elementary operators on Calkin Algebras. Irish Math Soc Bul.l 2001;46:33-42.
  7. Mathieu M. Elementary operators on prime C*-algebras. Irish Math Ann. 1989;284:223-244.
  8. Nyamwala FO, Agure JO. Norms of elementary operators in Banach algebras. Int Journal Math Anal. 2008;28:41-424.
  9. Okelo. NB, Agure JO, Ambogo DO. Norms of elementary operators and characterization of Norm-Attainable operators. Int Journal Math Anal. 2010;4:1197-1204.
  10. Seddik A. Rank one operators and norm of elementary operators, Linear Algebra and its Applications. 2007;424:177-183.
  11. Stacho LL, Zalar B. On the norm of Jordan elementary operators in standard algebras. Publ Math Debrecen. 1996;49:127-134.
  12. Timoney RM. Norms of elementary operators. Irish Math Soc Bull. 2001;46:        13-17.
  13. Vijayabalaji S, Shyamsundar G.  Interval-valued intuitionistic fuzzy transition matrices. Int J Mod Sci Technol. 2016;1(2):47-51.
  14. Judith J O,  Okelo NB, Roy K, Onyango T. Numerical Solutions of Mathematical Model on Effects of Biological Control on Cereal Aphid Population Dynamics. Int J Mod Sci Technol. 2016;1(4)138-143​​.
  15. Judith JO,  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. 1(5):150-158​​.
  16. Vijayabalaji S, Sathiyaseelan N. Interval-Valued Product Fuzzy Soft Matrices and its Application in Decision Making. Int J Mod Sci Technol 2016;1(7)159-163​​.
  17. Chinnadurai V,  Bharathivelan K. Cubic Ideals in Near Subtraction Semigroups. Int J Mod Sci Technol. 2016;1(8):276-282.
  18. Okello BO, Okelo NB, Ongati O. Characterization of Norm Inequalities for Elementary Operators. Int J Mod Sci Technol. 2017;2(3):81-84
  19. Wafula AM, Okelo NB, Ongati O. Norms of Normally Represented Elementary Operators. Int J Mod Sci Technol. 2018;3(1):10-16.​

International Journal of Modern Science and Technology

INDEXED IN 

ISSN 2456-0235