Norm Inequalities for Commutators of Normal Operators

  • Fuad Kittaneh
Conference paper
Part of the International Series of Numerical Mathematics book series (ISNM, volume 157)

Abstract

Let S, T, and X be bounded linear operators on a Hilbert space. It is shown that if S and T are normal with the Cartesian decompositions S = A+iC and T = B+iD such that a1Aa2, b1Bb2, c1Cc2, and d1Dd2 for some real numbers a1, a2, b1, b2, c1, c2, d1, and d2, then for every unitarily invariant norm |||·|||, and
$$ \left\| {ST - TS} \right\| \leqslant \frac{1} {2}\sqrt {(a_2 - a_1 )^2 + (c_2 - c_1 )^2 } \sqrt {(b_2 - b_1 )^2 + (d_2 - d_1 )^2 } , $$
where ‖·‖ is the usual operator norm. Applications of these norm inequalities are given, and generalizations of these inequalities to a larger class of nonnormal operators are also obtained.

Keywords

Commutator normal operator positive operator unitarily invariant norm norm inequality Cartesian decomposition 

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References

  1. [1]
    R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997.Google Scholar
  2. [2]
    R. Bhatia and F. Kittaneh, On the singular values of a product of operators, SIAM J. Matrix Anal. Appl. 11 (1990), 272–277.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    R. Bhatia and F. Kittaneh, Commutators, pinchings, and spectral variation, Oper. Matrices 2 (2008), 143–151.MATHMathSciNetGoogle Scholar
  4. [4]
    A. Böttcher and D. Wenzel, How big can the commutator of two matrices be and how big is it typically? Linear Algebra Appl. 403 (2005), 216–228.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    I.C. Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monographs 18, Amer. Math. Soc., Providence, RI, 1969.Google Scholar
  6. [6]
    F. Kittaneh, Norm inequalities for sums and differences of positive operators, Linear Algebra Appl. 383 (2004), 85–91.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    F. Kittaneh, Inequalities for commutators of positive operators, J. Funct. Anal. 250 (2007), 132–143.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    F. Kittaneh, Norm inequalities for commutators of positive operators and applications, Math. Z. 258 (2008), 845–849.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    A. McIntosh, Heinz inequalities and perturbation of spectral families, Macquarie Mathematical Reports 79-006 (1979).Google Scholar
  10. [10]
    J.G. Stampfli, The norm of a derivation, Pacific J. Math. 33 (1970), 737–747.MATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Fuad Kittaneh
    • 1
  1. 1.Department of MathematicsUniversity of JordanAmmanJordan

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