On Numerical Range of Multiplication Operator

Abstract

Let H be an in nite dimensional complex Hilbert space and A;B 2 B(H) where B(H) is the C -algebra of all bounded linear operators on H. Let MAB : B(H) ! B(H) be a multiplication operator induced by A and B de- ned by MAB(X) = AXB In this paper we show that the numerical range of multiplication operator is given by V (MAB=B(B(H))) = [ S U2U(H)W(U AUB)􀀀]􀀀 for all A;B 2 B(H) and U a unitary operator on the algebra B(H) where V is the algebraic nu- merical range and W is the classical numerical range. The results obtained are an extension of the the work done by Barraa [4]. Mathematics Subject Classi cation: Primary 47A12, Secondary 47A30, 47B47