The normed finiteness property of compact contraction operators |
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Authors: | Yuan-Chuan Li |
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Institution: | a Department of Applied Mathematics, National Chung-Hsing University, Taichung 402, Taiwan b Department of Mathematics, National Taiwan Normal University, 88 Sec. 4, Ting Chou Road, Taipei 116, Taiwan |
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Abstract: | We study the joint spectral radius given by a finite set of compact operators on a Hilbert space. It is shown that the normed finiteness property holds in this case, that is, if all the compact operators are contractions and the joint spectral radius is equal to 1 then there exists a finite product that has a spectral radius equal to 1. We prove an additional statement in that the requirement that the joint spectral radius be equal to 1 can be relaxed to the asking that the maximum norm of finite products of a length norm is equal to 1. The length of this product is related to the dimension of the subspace on which the set of operators is norm preserving. |
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Keywords: | Primary 15A18 15A48 15A60 47A30 |
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