On the Numerical Range of Compact Operators on Hilbert Spaces

Proof. If I is a cluster point of W(T), then there exists a sequence {(Tx,, x,)}, where Ix,ll = 1 for all n, converging to A. Since the unit ball in a Hilbert space is weakly sequentially compact, there exists a subsequence {s,,} which is weakly convergent to an x where 11x1 < 1. Since T is a compact operator, {Tx,,} is strongly convergent to Tx. However, I(Tx,,, x,,) (Tx, x)l < I( Tx,,, x,,) (Tx7 x,,>l + l (Tx, x,,) (Tx, x)l < Ilx,,I I I Tx,,Txll + I(x,,, Tx) ( 4 Tx)l. Therefore {(Tx,,, x,,)} converges to (Tx, x) and so (Tx, x) = A. If I # 0, clearly .x # 0, SO