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arXiv · 1404.5822

The spectrum of the product of operators, and the product of their numerical ranges

Chi-Kwong Li·Ming-Cheng Tsai·Kuo-Zhong Wang·Ngai-Ching Wong
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Abstract

We show that a compact operator $A$ is a multiple of a positive semi-definite operator if and only if $$ σ(AB) \subseteq \overline{W(A)W(B)}, \quad\text{for all (rank one) operators $B$}. $$ An example of a normal operator is given to show that the equivalence conditions may fail in general. We then obtain conditions to identify other classes of operators $A$ so that equivalence conditions hold.

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Chi-Kwong Li, Ming-Cheng Tsai, Kuo-Zhong Wang, Ngai-Ching Wong. 2014-07-12. The spectrum of the product of operators, and the product of their numerical ranges. https://arxiv.org/abs/1404.5822

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