DOI QR코드

DOI QR Code

Weighted Geometric Means of Positive Operators

Izumino, Saichi;Nakamura, Noboru

  • Received : 2009.09.16
  • Accepted : 2010.01.27
  • Published : 2010.06.30

Abstract

A weighted version of the geometric mean of k ($\geq\;3$) positive invertible operators is given. For operators $A_1,{\ldots},A_k$ and for nonnegative numbers ${\alpha}_1,\ldots,{\alpha}_k$ such that $\sum_\limits_{i=1}^k\;\alpha_i=1$, we define weighted geometric means of two types, the first type by a direct construction through symmetrization procedure, and the second type by an indirect construction through the non-weighted (or uniformly weighted) geometric mean. Both of them reduce to $A_1^{\alpha_1}{\cdots}A_k^{{\alpha}_k}$ if $A_1,{\ldots},A_k$ commute with each other. The first type does not have the property of permutation invariance, but satisfies a weaker one with respect to permutation invariance. The second type has the property of permutation invariance. We also show a reverse inequality for the arithmetic-geometric mean inequality of the weighted version.

Keywords

positive operator;weighted geometric mean;arithmetic-geometric mean inequality;reverse inequality

References

  1. E. Ahn, S. Kim, H. Lee and Y. Lim, Sagae-Tanabe weighted means and reverse in- equalities, Kyungpook Math. J., 47(2007), 595-600.
  2. T. Ando, C.-K. Li and R. Mathias, Geometric means, Linear Algebra Appl., 385(2004), 305-334. https://doi.org/10.1016/j.laa.2003.11.019
  3. E. Andruchow, G. Corach and D. Stojanoff, Geometrical significance of the Lowner- Heinz inequality, Proc. Amer. Math. Soc., 128(1999), 1031-1037.
  4. J. E. Cohen and R. D. Nussbaum, The arithmetic-geometric mean and its general- izations for noncommuting linear operators, Annali della Scuola Normale Sup di Pia Cl. Sci, (4) 15(1988) no. 2, 239-308.
  5. B. Q. Feng and A. Tonge, Geometric means and Hadamard products, Math. InequaI- ities Appl., 8(2005), 559-564.
  6. J.I. Fujii, M. Fujii, M. Nakamura, J. Pecaric and Y. Seo, A reverse inequality for the weighted geometric mean due to Lawson-Lim, Linear Algebra Appl., 427(2007), 272-284. https://doi.org/10.1016/j.laa.2007.07.025
  7. J.I. Fujii, M. Nakamura, J. Pecaric and Y. Seo, Bounds for the ratio and difference between parallel sum and series via Mond-Pecaric method, Math. Inequalities and Appl., 9(2006), 749-759.
  8. T. Furuta, J. Micic, J. Pecaric and Y. Seo, Mond-Pecaric Method in Operator In- equalities, Monographs in Inequalities I, Element, Zagreb, 2005.
  9. S. Izumino and N. Nakamura, Geometric means of positive operators, II, Sci. Math.Japon., 69(2009), 35-44.
  10. S. Kim and Y. Lim, A converse inequality of higher order weighted arithmetic and geometric means of positive definite operators, Linear Alg. Appl., 426(2007), 490-496. https://doi.org/10.1016/j.laa.2007.05.028
  11. H. Kosaki, Geometric mean of several positive operators, 1984.
  12. F. Kubo, and T. Ando, Means of positive linear operators, Math. Ann., 246(1980), 205-224. https://doi.org/10.1007/BF01371042
  13. J. Lawson and Y. Lim, A general framework for extending means to higher orders, Colloq. Math., 113(2008), 191-221. https://doi.org/10.4064/cm113-2-3
  14. N. Nakamura, Geometric means of positive operators, Kyungpook Math. J., 49(2009), 167-181. https://doi.org/10.5666/KMJ.2009.49.1.167
  15. M. Sagae and K. Tanabe, Upper and lower bounds for the arithmetic-geometric- harmonic means of positive definite matrices, Linear and Multilinear Alg., 37(1994), 279-282. https://doi.org/10.1080/03081089408818331
  16. T. Yamazaki, An extension of Kantorovich inequality to n-operators via the geometric mean by Ando-Li-Mathias, Linear Algebra Appl., 416(2006), 688-695. https://doi.org/10.1016/j.laa.2005.12.013