An application of grand Furuta inequality to a type of operator equation

  • Authors

    • Jian Shi Hebei University
    2014-09-19
    https://doi.org/10.14419/gjma.v2i4.3400

  • The existence of positive semidefinite solutions ofthe operator equation $\displaystyle\sum_{j=1}^{n}A^{n-j}XA^{j-1}=Y$ is investigated by applying grand Furuta inequality. If there  exists positive semidefinite solutions of the operator equation, one of the special types of Y is obtained, which extends the related result before. Finally, an example is given based on our result.

    Keywords: grand Furuta inequality, operator equation, matrix equation, positive semidefinite operator.

  • References

    1. T. Ando, On some operator inequalities, Math. Ann. 279 (1987), 157-159.
    2. T. Ando and F. Hiai, Log majorization and complementary Golden-Thompson type inequalities, Linera Algebra Appl.197 (1994), 113-131.
    3. A. Aluthge, On p-hyponormal operators for 0 < p < 1, Integr. Equat. Oper. Th. 13 (1990), 307-315.
    4. R. Bhatia and M. Uchiyama, The operator equation sum^{n}_{i=0}A^{n-i}XB^{i}=Y, Expo. Math. 27 (2009), 251-255.
    5. M. Fujii, T. Furuta and E. Kamei, Furuta's inequality and its application to Ando's theorem, Linear Algebra Appl. 179 (1993), 161-169.
    6. M. Fujii and E. Kamei, Mean theoretic approach to the grand Furuta inequality, Proc. Amer. Math. Soc. 124 (1996), 2751-2756.
    7. M. Fujii, A. Matsumoto and R. Nakamoto, A short proof of the best possibility for the grand Furuta inequality, J. Inequal. Appl. 4 (1999), 339-344.
    8. T. Furuta, Ageqslant Bgeqslant 0 assures (B^{r}A^{p}B^{r})^{1/q}geqslant B^{(p+2r)/q} for rgeqslant 0, pgeqslant 0, qgeqslant 1 with (1+2r)qgeqslant p+2r, Proc. Amer. Math. Soc. 101 (1987), 85-88.
    9. T. Furuta, An extension of the Furuta inequality and Ando-Hiai log-majorization, Linear Algebra Appl. 219 (1995), 139-155.
    10. T. Furuta, Simplied proof of an order preserving operator inequality, Proc. Japan Acad. 74 (1998), 114.
    11. T. Furuta and M. Yanagida, On powers of p-hyponormal and log-hyponormal operators, J. Inequal. Appl. 5 (2000), 367-380.
    12. T. Furuta, Positive semidefinite solutions of the operator equation sum_{j=1}^{n}A^{n-j}XA^{j-1}=B, Linear Algebra Appl. 432 (2010), 949-955.
    13. E. Heinz, Beiträge zur Störungsteorie der Spektralzerlegung, Math. Ann. 123 (1951), 415-438.
    14. M. Ito and T. Yamazaki, Relations between two inequalities (B^{frac r 2}A^{p}B^{frac r 2})^{frac {r}{p+r}}geq B^{r} and (A^{frac r 2}B^{p}A^{frac r 2})^{frac {p}{p+r}}geq A^{r} and their applications, Integr. Equat. Oper. Th. 44 (2002), 442-450.
    15. C.-S. Lin, On operator order and chaotic operator order for two operators, Linear Algebra Appl. 425 (2007), 1-6.
    16. K. Löwner, Über monotone MatrixFunktionen, Math. Z. 38 (1934), 177-216.
    17. K. Tanahashi, Best possibility of the Furuta inequality, Proc. Amer. Math. Soc. 124 (1996), 141-146.
    18. K. Tanahashi, The best possibility of the grand Furuta inequality, Proc. Amer. Math. Soc. 128 (2000), 511-519.
    19. T. Yamazaki, Simplified proof of Tanahashi's result on the best possibility of generalized Furuta inequality, Math. Inequal. Appl. 2 (1999), 473-477.
    20. C. Yang and H. Dai, An application of Furuta inequality and its best possibility, Applied Mathematics, A Journal of Chinese Universities Series B. 23 (3)(2008), 326-330.
    21. C. Yang and J. Yuan, Extensions of the results on powers of p-hyponormal and log-hyponormal operators, J. Inequal. Appl. 1 (2006), 1-14.
    22. J. Yuan and Z. Gao, Structure on powers of p-hyponormal and log-hyponormal operators, Integr. Equat. Oper. Th. 59 (2007), 437-448.
    23. J. Yuan and Z. Gao, Classified construction of generalized Furuta type operator functions, Math. Inequal. Appl. 11 (2008), 189-202.
    24. J. Yuan and Z. Gao, Complete form of Furuta inequality, Proc. Amer. Math. Soc. 136 (2008), 2859-2867.
    25. J. Yuan, Classified construction of generalized Furuta type operator functions, II, Math. Inequal. Appl. 13 (2010), 775-784.
    26. J. Yuan, Furuta inequality and q-hyponormal operators, Oper. Matrices 4 (2010), 405-415.

  • Downloads

  • How to Cite

    Shi, J. (2014). An application of grand Furuta inequality to a type of operator equation. Global Journal of Mathematical Analysis, 2(4), 281-285. https://doi.org/10.14419/gjma.v2i4.3400