Advertisement

Positivity

, Volume 10, Issue 2, pp 251–260 | Cite as

Norm Inequalities for Sums of Positive Operators. II

  • Fuad Kittaneh
Article

Abstract

It is shown that if A and B are positive operators on a separable complex Hilbert space, then

Open image in new window

for every unitarily invariant norm. When specialized to the usual operator norm ||·|| and the Schatten p-norms ||·|| p , this inequality asserts that

Open image in new window

and

Open image in new window

These inequalities improve upon some earlier related inequalities. Other norm inequalities for sums of positive operators are also obtained.

Mathematics Subject Classification 2000

47A30 47B10 47B15 

Keywords

Operator matrix positive operator Schatten p-norm unitarily invariant norm norm inequalitys 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bhatia, R.: Matrix Analysis, Springer-Verlag, New York, 1997.Google Scholar
  2. 2.
    Bhatia, R. and Kittaneh, F.: Norm inequalities for partitioned operators and an application, Math. Ann. 287 (1990), 719–726.Google Scholar
  3. 3.
    Bhatia, R. and Kittaneh, F.: Clarkson inequalities with several operators, Bull. London Math. Soc., 36 (2004), 820–832.Google Scholar
  4. 4.
    Davidson, K. and Power, S.C.: Best approximation in C*-algebras, J. Reine Angew. Math. 368 (1986), 43–62.Google Scholar
  5. 5.
    Gohberg, I.C. and Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monographs, Vol. 18, Amer. Math. Soc., Providence, RI, 1969.Google Scholar
  6. 6.
    Horn, R.A. and Zhan, X.: Inequalities for C-S seminorms and Lieb functions, Linear Algebra Appl. 291 (1999), 103–113.Google Scholar
  7. 7.
    King, C.: Inequalities for trace norms of 2 × 2 block matrices, Comm. Math. Phys. 242 (2003), 531–545.Google Scholar
  8. 8.
    Kittaneh, F.: On the continuity of the absolute value map in the Schatten classes, Linear Algebra Appl. 118 (1989), 61–68.Google Scholar
  9. 9.
    Kittaneh, F.: Norm inequalities for certain operator sums, J. Funct. Anal. 103 (1997), 337–348.Google Scholar
  10. 10.
    Kittaneh, F.: Norm inequalities for sums of positive operators, J. Operator Theory 48 (2002), 95–103.Google Scholar
  11. 11.
    Kittaneh, F.: Norm inequalities for sums and differences of positive operators, Linear Algebra Appl. 383 (2004), 85–91.Google Scholar
  12. 12.
    Simon, B.: Trace Ideals and Their Applications, Cambridge University Press, Cambridge, 1979.Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of JordanAmmanJordan

Personalised recommendations