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

By Chi-Kwong Li, M. Tsai, Kuo-Zhong Wang, Ngai-Ching Wong. 2014

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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 ( A B ) ⊆ W ( A ) W ( B ) ¯ , 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.

Subfields: Algebra and Number Theory, Applied Mathematics, Computational Theory and Mathematics