المجلة الدولية للعلوم والتقنية

P.R. Halmos [3] has proved that. For any bounded operator T and a Hilbert space H, every eigen value  of T is in the numerical range W(T), and the converse is true if ||=ווT וו. In this paper we generalize the above result to the case of n-tuples of the operators on any normed linear space. (semi-inner product space) with some necessary condition for the converse. Further, we improve a theorem of lumer [5] which asserts that : the approximate point spectrum (T) of T B(X), where X is any normed space is contained in (W(T)) ̅ where dash is denotes closure.