Vol. 32, No. 3, 1970

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The numerical range of an operator

Mary Rodriguez Embry

Vol. 32 (1970), No. 3, 647–650

Let A be a continuous linear operator on a complex Hilbert space X with inner product <,> and associated norm ∥∥. Let W(A) = {⟨Ax,x⟩∥|x= 1} be the numerical range of A and for each complex number z let Mz = {x|⟨Ax,x= zx2}. Let Y Mz be the linear span of Mz and Mz Mz = {x + y|x Mz and y Mz}. An element z of W(A) is characterized in terms of the set Mz as follows: THEOREM 1. If z W(A), then Y Mz = Mz Mz and (i) z is an extreme point of W(A) if and only if Mz is linear; (ii) if z is a nonextreme boundary point of W(A), then Y Mz is a closed linear subspace of X and Y Mz = ∪{Mw|w L}, where L is the line of support of W(A), passing through z. In this case Y Mz = X if and only if W(A) L. (iii) if W(A) is a convex body, then x is an interior point of W(A) if and only if Y Mz = X.

Mathematical Subject Classification
Primary: 47.10
Received: 14 May 1969
Published: 1 March 1970
Mary Rodriguez Embry