Anderson’s theorem for compact operators

Gau, Hwa-Long; Wu, Pei

doi:10.1090/S0002-9939-06-08699-0

Anderson’s theorem for compact operators
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by Hwa-Long Gau and Pei Yuan Wu PDF

Proc. Amer. Math. Soc. 134 (2006), 3159-3162 Request permission

Abstract:

It is shown that if $A$ is a compact operator on a Hilbert space with its numerical range $W(A)$ contained in the closed unit disc $\overline {\mathbb {D}}$ and with $\overline {W(A)}$ intersecting the unit circle at infinitely many points, then $W(A)$ is equal to $\overline {\mathbb {D}}$. This is an infinite-dimensional analogue of a result of Anderson for finite matrices.

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