On the Numerical Ranges

Abstract

Properties of operators in a C*-algebra B(H) have been studied by many researchers in operator theory. This paper is an investigation of the numerical ranges of operators in B(H) . We show that zero is in the algebraic numerical range of an operator in B(H) if and only if that operator is orthogonal to the identity operator. We then show that the algebraic numerical range of an operator in B(H) is convex and is also equal to the closure of the spatial numerical range of that operator. We employ the inner products of vectors in a Hilbert space H as well as the properties of the states in B(H) in obtaining our results.