Inequalities between ∥f(A + B)∥ and ∥f(A) + f(B)∥ |
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Authors: | Toma? Kosem |
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Institution: | Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia |
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Abstract: | The conjecture posed by Aujla and Silva J.S. Aujla, F.C. Silva, Weak majorization inequalities and convex functions, Linear Algebra Appl. 369 (2003) 217-233] is proved. It is shown that for any m-tuple of positive-semidefinite n × n complex matrices Aj and for any non-negative convex function f on 0, ∞) with f(0) = 0 the inequality ?f(A1) + f(A2) + ? + f(Am)? ? ? f(A1 + A2 + ? + Am)? holds for any unitarily invariant norm ? · ?. It is also proved that ?f(A1) + f(A2) + ? + f(Am)? ? f(?A1 + A2 + ? + Am?), where f is a non-negative concave function on 0, ∞) and ? · ? is normalized. |
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Keywords: | 15A60 47A30 |
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