International Journal of Modern Computation, Information and Communication Technology

ISSN 2581-5954

March-April 2021, Vol. 4, Issue 3-4, p. 7-12.​​

Numerical radii inequalities for derivations induced by orthogonal projections
I. O. Okwany, N. B. Okelo*, O. 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:


Let Hn be a finite dimensional Hilbert space and δP,Q be a generalized derivation induced by the orthogonal projections P and Q. In this study, authors have approximated the norm of δP,Q and showed that δP,Q is bounded and is utmost equal to the sum of norms of P and Q. Authors have also considered the linearity and inequalities of in the context of tensor products. Finally, authors have determined the Hilbert-Schmidt and completely bounded norm inequalities for δP,Q.

Keywords: Hilbert space; Orthogonal projections; Hilbert-Schmidt; Inequalities.


  1. Anderson JH, Foias C. Properties which normal operators share with normal derivation and related operators. Pacific J. Math. 1976;61:313-25.
  2. Archbold RJ. On the norm of an inner derivation of a algebra. Math. Proc. Camb. Phil. Soc.  1978;84(2):273-91.
  3. Archbold RJ, Eberhard KB, Sommerset D.W.B.  Norms of inner derivations for multiplier algebras of algebras and groups of algebras. Journal of Functional Analysis 2012;262:2050-73.
  4. Bhatia R, Kittaneh F. Inequalities for commutators of positive operators. J. Funct. Anal. 2007;250:132-43.
  5. Bonyo JO, Agure JO. Norms of Inner Derivations on Norm ideals. Int. J. Math. Anal. 2010;4(14):695-701.
  6. Bonyo JO, Agure JO. Norms of derivations implemented by -universal operators. Int. Math. forum 2011;5(5):215-22.
  7. Gajendragadkar P. Norm of derivation on a von Neumann algebra (′). Transactions of the American Mathematical Society 1972;170;165-70.
  8. Johnson BE. Norms of derivations on , Pacific Journal of Mathematics 1971;38(2):465-9.
  9. Kaplansky IM. Modules over operators algebras. Amer. J. Math. 1953;75:839-58.
  10. Kittaneh F. Linear Algebra and its Applications, Elsevier Science Inc., New York, 1994.
  11. Li CK, Woerdeman HA.  Problem on the (1, k)-numerical radius. Linear and Multilinear Algebra 2019;22(3):739-50.
  12. Li CK, Tsing NK, Uhling F. Numerical range of an operator on an indefinite inner product space. International Linear Algebra Society 1996;1:1-17.
  13. Matej B. Characterizations of Derivations on Some Normed Algebras with Involution. Journal of Algebra 1992;152:454-62.
  14. Mohsen EO, Mohammad SM, Niknam A. Some numerical radius inequalities for Hilbert Space operators. Journal of Inequalities and Applications 2009;2(4):471-8.
  15. Okelo NB, Agure JO, Ambogo DO. Norms of Elementary Operator and Characterizaion of Norm-attainable operators. Int. J. Math. Anal. 2010;4(24):1197-204.
  16. Okelo NB, Agure JO, Oleche PO. Certain conditions for norm-attainability of elementary operators and derivations. Int. J. Math. Soft Computing 2013;3(1):53-9.
  17. Ringrose JR. Automatic continuity of derivations of operator algebras J. London Math. Soc. 1072;5(2):432.
  18. Rosenblum M. On the operator equation BX-ZA=Q. Duke Math. J. 1956;23:263-9.
  19. Salah M. Some recent results on operator commutators and related operators with applications. Journal of Mathematical Inequalities 2019;13:1129-1146.
  20. Spivack M., Derivations on commutative operator algebras, Bull. Austral. Math. Soc. 1985;32:415-18.
  21. Stampfli JG. A norm of a derivation, Pacific Journal of Mathematics 1970;33(3): 737-47.
  22. Vassili G. Functional Analysis I, Springer Verlag, Berlin 2011.
  23. Weber RE. Derivations And The Trace-Class Operators. Proc. Amer. Math. Soc. 1979;3(1):79-82.