International Mathematical Forum, Vol. 12, 2017, no. 6, 277 - 283 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.611148 On the Numerical Range of a Generalized Derivation F. M. Runji Department of Mathematics Statistics and Actuarial Science Karatina University, Karatina, Kenya J. O. Agure Department of Pure and Applied Mathematics Maseno University, Maseno, Kenya F. O. Nyamwala Department of Physics and Mathematics Moi University, Eldoret, Kenya Copyright c  2016 F. M. Runji, J. O. Agure and F. O. Nyamwala. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We examine the relationship between the numerical range of the restriction of a generalized derivation to a norm ideal J and that of its implementing elements. Mathematics Subject Classification: 47A12; 47B47 Keywords: Numerical range, Generalized Derivation 1 Introduction Given a Banach algebra A , A * the dual of A , S (A )= {x ∈ A : ‖x‖ =1}, the unit sphere, and x ∈ S (A ), let D(x, A )= {f ∈ A * : f(x) =1= ‖f‖}.