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UNITARILY INVARIANT NORM INEQUALITIES INVOLVING G1 OPERATORS

  • Bakherad, Mojtaba (Department of Mathematics Faculty of Mathematics University of Sistan and Baluchestan)
  • Received : 2017.08.05
  • Accepted : 2017.12.21
  • Published : 2018.07.31

Abstract

In this paper, we present some upper bounds for unitarily invariant norms inequalities. Among other inequalities, we show some upper bounds for the Hilbert-Schmidt norm. In particular, we prove $${\parallel}f(A)Xg(B){\pm}g(B)Xf(A){\parallel}_2{\leq}{\Large{\parallel}}{\frac{(I+{\mid}A{\mid})X(I+{\mid}B{\mid})+(I+{\mid}B{\mid})X(I+{\mid}A{\mid})}{^dA^dB}}{\Large{\parallel}}_2$$, where A, B, $X{\in}{\mathbb{M}}_n$ such that A, B are Hermitian with ${\sigma}(A){\cup}{\sigma}(B){\subset}{\mathbb{D}}$ and f, g are analytic on the complex unit disk ${\mathbb{D}}$, g(0) = f(0) = 1, Re(f) > 0 and Re(g) > 0.

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References

  1. R. Alizadeh and M. B. Asadi, An extension of Ky Fan's dominance theorem, Banach J. Math. Anal. 6 (2012), no. 1, 139-146. https://doi.org/10.15352/bjma/1337014672
  2. T. Ando and X. Zhan, Norm inequalities related to operator monotone functions, Math. Ann. 315 (1999), no. 4, 771-780. https://doi.org/10.1007/s002080050335
  3. M. Bakherad and F. Kittaneh, Numerical radius inequalities involving commutators of $G_1$ operators, Complex Anal. Oper. Theory 2017 (2017). https://doi.org/10.1007/s11785-017-0726-9.
  4. M. Bakherad and M. S. Moslehian, Reverses and variations of Heinz inequality, Linear Multilinear Algebra 63 (2015), no. 10, 1972-1980. https://doi.org/10.1080/03081087.2014.880433
  5. L. Bao, Y. Lin, and Y. Wei, Krylov subspace methods for the generalized Sylvester equation, Appl. Math. Comput. 175 (2006), no. 1, 557-573. https://doi.org/10.1016/j.amc.2005.07.041
  6. R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics, 169, Springer-Verlag, New York, 1997.
  7. W. F. Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation, Springer-Verlag, New York, 1974.
  8. N. Dunford and J. T. Schwartz, Linear Operators. Part II, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963.
  9. J. I. Fujii, M. Fujii, T. Furuta, and M. Nakamoto, Norm inequalities equivalent to Heinz inequality, Proc. Amer. Math. Soc. 118 (1993), no. 3, 827-830. https://doi.org/10.1090/S0002-9939-1993-1132412-1
  10. S. Hildebrandt, Uber den numerischen Wertebereich eines Operators, Math. Ann. 163 (1966), 230-247. https://doi.org/10.1007/BF02052287
  11. O. Hirzallah and F. Kittaneh, Singular values, norms, and commutators, Linear Algebra Appl. 432 (2010), no. 5, 1322-1336. https://doi.org/10.1016/j.laa.2009.10.042
  12. O. Hirzallah and F. Kittaneh, Norm inequalities for commutators of $G_1$ operators, Complex Anal. Oper. Theory 10 (2016), no. 1, 109-114. https://doi.org/10.1007/s11785-015-0485-4
  13. F. Kittaneh, M. S. Moslehian, and M. Sababheh, Unitarily invariant norm inequalities for elementary operators involving $G_1$ operators, Linear Algebra Appl. 513 (2017), 84-95. https://doi.org/10.1016/j.laa.2016.10.010
  14. F. Kittaneh, M. S. Moslehian, and T. Yamazaki, Cartesian decomposition and numerical radius inequalities, Linear Algebra Appl. 471 (2015), 46-53. https://doi.org/10.1016/j.laa.2014.12.016
  15. C. R. Putnam, Operators satisfying a $G_1$ condition, Pacific J. Math. 84 (1979), no. 2, 413-426. https://doi.org/10.2140/pjm.1979.84.413
  16. A. Seddik, Rank one operators and norm of elementary operators, Linear Algebra Appl. 424 (2007), no. 1, 177-183. https://doi.org/10.1016/j.laa.2006.10.003
  17. J. L. van Hemmen and T. Ando, An inequality for trace ideals, Comm. Math. Phys. 76 (1980), no. 2, 143-148. https://doi.org/10.1007/BF01212822