Numerical ranges of cube roots of the identity |
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Authors: | Thomas Ryan Harris Michael Mazzella David Renfrew Ilya M. Spitkovsky |
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Affiliation: | a Mathematics Department, California Polytechnic State University, San Luis Obispo, CA 93407, United States b Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, United States |
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Abstract: | The numerical range of a bounded linear operator T on a Hilbert space H is defined to be the subset W(T)={〈Tv,v〉:v∈H,∥v∥=1} of the complex plane. For operators on a finite-dimensional Hilbert space, it is known that if W(T) is a circular disk then the center of the disk must be a multiple eigenvalue of T. In particular, if T has minimal polynomial z3-1, then W(T) cannot be a circular disk. In this paper we show that this is no longer the case when H is infinite dimensional. The collection of 3×3 matrices with three-fold symmetry about the origin are also classified. |
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Keywords: | Numerical range Algebraic operator Threefold symmetry |
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