Norm inequalities for commutators
University of Macau === Faculty of Science and Technology === Department of Mathematics
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University of Macau
2010
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ndltd-MACAU-oai-libdigital.umac.mo-b21828772013-01-07T23:06:55Z2010http://umaclib3.umac.mo/record=b2182877UM_THESESUniversity of MacauFaculty of Science and TechnologyDepartment of MathematicsUniversity of MacauUniversity of Macau -- Dissertations澳門大學 -- 論文Commutators (Operator theory)Matrices -- NormsMathematics -- Department of MathematicsengFong, Kin SioNorm inequalities for commutatorsIt has been conjectured and proved that ||XY – YX|| <sub>F</sub> ≤ √2||X||<sub>F</sub> ||Y||<sub>F</sub> , for any n x n complex matrices X and Y , where ||‧||<sub>F</sub> denotes the Frobenius norm. In this thesis, we show that the commutator XY - Y X in the above inequality can be replaced by the product XY - Y X<sup>T</sup>for real matrices X and Y , where X<sup>T</sup> denotes the transpose of X. The proof is given in Chapter 2. We also give the characterization of those pairs of matrices that satisfy the inequality with equality in Chapter 3. Audenaert showed that for any n x n complex matrices X and Y , the above inequality can be strengthened as ||XY – YX||<sub>F</sub> ≤ √2||X||<sub>F</sub> ||Y||<sub>(2),2,</sub> where ||‧||<sub>(2),2,</sub> denotes the (2, 2)-norm. In Chapter 4 we show that the commutator XY - YX in this inequality can also be replaced by the product XY - YX<sup>T</sup> for real matrices X and Y . Those pairs of matrices which satisfy the inequality with equality are also characterized. It has been conjectured and proved that ||XY – YX|| <sub>F</sub> ≤ √2||X||<sub>F</sub> ||Y||<sub>F</sub> for any n x n complex matrices X and Y , where ||‧||<sub>F</sub> denotes the Frobenius norm. A characterization of those pairs of matrices that satisfy the inequality with equality has also been found. Thereafter, Audenaert gave another proof for the inequality by means of what he called the matrix version of variance. Based on his proof, we find another proof for the equality cases in Chapter 2. Audenaert also showed that ||XY – YX||<sub>F</sub> ≤ √2||X||<sub>F</sub> ||Y||<sub>(2),2,</sub> where ||‧||<sub>(2),2, denotes the (2,2)-norm. In Chapter 3 we characterize the pairs of matrices which satisfy the inequality with equality. Furthermore, we extend this inequality to other Schatten p-norms in Chapter 4. On the other hand, Bottcher and Wenzel proved that for any unitarily invariant norm ||‧||, sup{||XY-YX|| / ||X||||Y||:X and Y are n x n nonzero complex matrices) = C≥√2 They also asked whether the Frobenius norm is the only one having such property. In Chapter 5 we answer the question by showing that the dual norm of the (2, 2)- norm also has the property that C =√2. |
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NDLTD |
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English |
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University of Macau -- Dissertations 澳門大學 -- 論文 Commutators (Operator theory) Matrices -- Norms Mathematics -- Department of Mathematics |
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University of Macau -- Dissertations 澳門大學 -- 論文 Commutators (Operator theory) Matrices -- Norms Mathematics -- Department of Mathematics Fong, Kin Sio Norm inequalities for commutators |
| description |
University of Macau === Faculty of Science and Technology === Department of Mathematics |
| author |
Fong, Kin Sio |
| author_facet |
Fong, Kin Sio |
| author_sort |
Fong, Kin Sio |
| title |
Norm inequalities for commutators |
| title_short |
Norm inequalities for commutators |
| title_full |
Norm inequalities for commutators |
| title_fullStr |
Norm inequalities for commutators |
| title_full_unstemmed |
Norm inequalities for commutators |
| title_sort |
norm inequalities for commutators |
| publisher |
University of Macau |
| publishDate |
2010 |
| url |
http://umaclib3.umac.mo/record=b2182877 |
| work_keys_str_mv |
AT fongkinsio norminequalitiesforcommutators |
| _version_ |
1716479060374192128 |