It is shown that if A A is a compact operator on a Hilbert space with its numerical range W ( A ) W(A) contained in the closed unit disc D ¯ D̄ and with W ( A ) ¯ W(A)̄ intersecting the unit circle at infinitely many points, then W ( A ) W(A) is equal to D ¯ D̄ . This is an infinite-dimensional analogue of a result of Anderson for finite matrices.
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Gau et al. (2006) studied this question.