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⟩ = z∥x∥2}. 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.
