On the existence of unitarily invariant norm under some conditions |
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Authors: | R Alizadeh GH Esslamzadeh |
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Institution: | 1. Faculty of Sciences, Department of Mathematics , Shahed University , Tehran, Iran alizadeh@shahed.ac.ir;3. Faculty of Sciences, Department of Mathematics , Shiraz University , Shiraz, Iran |
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Abstract: | Let r 1, …, r m be positive real numbers and A 1, …, A m be n × n matrices with complex entries. In this article, we present a necessary and sufficient condition for the existence of a unitarily invariant norm ‖·‖, such that ‖A i ‖ = r i , for i = 1, …, m. Then we identify the greatest unitarily invariant norm which satisfies this condition. Using this, we get an approximation of unitarily invariant norms. Although the minimum unitarily invariant norm which satisfies this condition does not exist in general, we find conditions over A i s and r i s which are sufficient for the existence of such a norm. Finally, we get a characterization of unitarily invariant norms. |
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Keywords: | unitarily invariant norm singular value Ky Fan norm |
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