UM ETheses Collection (澳門大學電子學位論文庫)
 Title

Norm inequalities for commutators
 English Abstract

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It has been conjectured and proved that kXY − Y XkF ≤ √ 2kXkF kY kF for any n × n complex matrices X and Y , where k · kF 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 kXY − Y XkF ≤ √ 2kXkF kY k(2),2, where k · k(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 pnorms in Chapter 4. On the other hand, B¨ottcher and Wenzel proved that for any unitarily invariant norm k · k, sup kXY − Y Xk kXkkY k : X and Y are n × 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.
 Issue date

2010.
 Author

Fong, Kin Sio
 Faculty

Faculty of Science and Technology
 Department

Department of Mathematics
 Degree

M.Sc.
 Subject

Commutators (Operator theory)
Matrices  Norms
 Supervisor

Cheng, Che Man
Leong, Ieng Tak
 Files In This Item
 Location
 1/F Zone C
 Library URL
 991005549199706306