On Numerical Range of Multiplication Operator

Abstract

Let H be an infinite dimensional complex Hilbert space and A, B ∈ 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 defined by MAB(X) = In this paper we show that the numerical range of multiplication operator is given by V (MAB/B(B(H))) = [S U∈U(H) W(U ∗AUB) −] − for all A, B ∈ B(H) and U a unitary operator on the algebra B(H) where V is the algebraic numerical range and W is the classical numerical range. The results obtained are an extension of the the work done by Barraa [4]. Mathematics Subject Classification: Primary 47A12, Secondary 47A30, 47B47