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International Journal of Modern Computation, Information and Communication Technology

ISSN 2581-5954

August-September 2019, Vol. 2, Issue 8-9, p. 59-65.​​

Derivation Properties of Finite Rank Operators
M. F. C. Kaunda, N. B. Okelo*, Omolo Ongati
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

In the present work, authors established derivation properties and range-kernel orthogonality of finite rank inner derivations implemented by finite rank hyponormal operators. The results show that an inner derivation is linear and bounded. Also by inner product trace and properties of adjoint, the inner derivation is self-adjoint if the inducing operator is self-adjoint. For orthogonality, we employ operator techniques such as properties of operators and derivation inequalities due to Anderson, Bouali, Maher, Mecheri and Halmos generalization formula to establish the orthogonality.

Keywords: Orthogonality; Hyponormal operators; Commutator; Finite rank and derivation.

References

  1. Akram MG. Best approximation and best co-approximation in normed spaces, thesis, Islamic university of Gaza; 2010.
  2. Alonso J, Benitez C. Orthogonality in normed linear spaces: a survey.I. Main properties. Extract Math 1988;3:1-15.
  3. Alonso J, Benitez C. Orthogonality in normed linear spaces: a survey.II. Relations between main orthogonalities. Extract Math 1989;4(3):121-31.
  4. Bachir A. Generalized derivation. SUT Journal of Math 2004;40(2):111-6.
  5. Bachir A. Range-kernel orthogonality of generalized derivations. Int J Math 2012;5(4):29-38.
  6. Benitez C. Orthogonality in normed linear spaces: a classification of the different concepts and some open problem, Universidad de Extremadura-Badajaz Spain; 2017.
  7. Berberian SK. A note on hyponormal operators. Pacific J Math 1962;12(206):1171-75.
  8. Bhuwan OP. Some new types of orthogonalities in normed spaces and application in best approximation, Journal of Advanced College of Engeneering and Management 2016;6:33-43.
  9. Birkhoff, G., Orthogonality in linear metric spaces. Duke Math J (1935);1:169-72.
  10. Bouali S, Bouhafsi Y. On the range- kernel orthogonality and p-symmetric operators. Math Ine Appl J 2006;9:511-19.
  11. Carlsson SO. Orthogonality in normed linear spaces, Ark Math 1962;4:297-318.
  12. Halmos PR. A Hilbert space problem book, Van Nostrand. Princeton; 1967.
  13. Hawthorne C. A brief introduction to trace class operators, Department of mathematics, University of Toronto; 2015.
  14. Okelo NB. Certain properties of Hilbert space operators, Int J Mod Sci Technol 2018;3(6):126-132.
  15. Okelo NB. Certain Aspects of Normal Classes of Hilbert Space Operators. Int J Mod Sci Technol 2018;3(10):203-207.
  16. Okelo NB. Characterization of Numbers using Methods of Staircase and Modified Detachment of Coefficients. International Journal of Modern Computation, Information and Communication Technology 2018;1(4):88-92.
  17. Okelo NB. On Characterization of Various Finite Subgroups of Abelian Groups. International Journal of Modern Computation, Information and Communication Technology 2018; 1(5):93-98.
  18. Okelo NB. On Normal Intersection Conjugacy Functions in Finite Groups. International Journal of Modern Computation, Information and Communication Technology 2018; 1(6):111-15.
  19. Okwany I, Odongo D, Okelo NB. Characterizations of Finite Semigroups of Multiple Operators. International Journal of Modern Computation, Information and Communication Technology 2018; 1(6):116-20.
  20. Ramesh R, Mariappan R. Generalized open sets in Hereditary Generalized Topological Spaces. J Math Comput Sci 2015;5(2):149-59.
  21. Saha S. Local connectedness in fuzzy setting. Simon Stevin 1987;61:3-13.