Weighted Geometric Means of Positive Operators
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
A weighted version of the geometric mean of k (≥ 3) positive invertible operators is given. For operators A1,...,Ak and for nonnegative numbers α1,...,αk such that Pk ∑ αi=1 we de¯ne weighted geometric means of two types, the ¯rst 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 A1α1...Akαk if A1,...,Ak commute with each other. The first type does not have the property of permutation invariance, but satis¯es 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. also show a reverse inequality for the arithmetic-geometric mean inequality of the weighted version.
Abstract
A weighted version of the geometric mean of k (≥ 3) positive invertible operators is given. For operators A1,...,Ak and for nonnegative numbers α1,...,αk such that Pk ∑ αi=1 we de¯ne weighted geometric means of two types, the ¯rst 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 A1α1...Akαk if A1,...,Ak commute with each other. The first type does not have the property of permutation invariance, but satis¯es 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. also show a reverse inequality for the arithmetic-geometric mean inequality of the weighted version.
참고문헌 (16)
- Eunkyung Ahn, "Sagae-Tanabe Weighted Means and Reverse Inequalities", Kyungpook Mathematical Journal, 47(4), pp.595~600 (2007), 경북대학교 자연과학대학 수학과
- T. Ando, "Geometric means", Linear Algebra Appl., 385, pp.305~334 (2004)
- E. Andruchow, "Geometrical significance of the Löwner-Heinz inequality", Proc. Amer. Math. Soc., 128, pp.1031~1037 (1999)
- J. E. Cohen, "The arithmetic-geometric mean and its general-izations for noncommuting linear operators", Annali della Scuola Normale Sup di Pia Cl. Sci, (4), 15(2), pp.239~308 (1988)
- B. Q. Feng, "Geometric means and Hadamard products", Math. Inequal-ities Appl., 8, pp.559~564 (2005)
- J.I. Fujii, "A reverse inequality for the weighted geometric mean due to Lawson-Lim", Linear Algebra Appl., 427, pp.272~284 (2007)
- J.I. Fujii, "Bounds for the ratio and difference between parallel sum and series via Mond-Pečarić method", Math. Inequalities and Appl., 9, pp.749~759 (2006)
- T. Furuta, "Mond-Pečarić Method in Operator In-equalities, Monographs in Inequalities I" (2005), Element
- S. Izumino, "Geometric means of positive operators, Ⅱ", Sci. Math. Japon., 69, pp.35~44 (2009)
- S. Kim, "A converse inequality of higher order weighted arithmetic and geometric means of positive definite operators", Linear Alg. Appl., 426, pp.490~496 (2007)
- H. Kosaki, "Geometric mean of several positive operators", ~ (1984)
- F. Kubo, "Means of positive linear operators", Math. Ann., 246, pp.205~224 (1980)
- J. Lawson, "A general framework for extending means to higher orders", Colloq. Math., 113, pp.191~221 (2008)
- Noboru Nakamura, "Geometric Means of Positive Operators", Kyungpook Mathematical Journal, 49(1), pp.167~181 (2009), 경북대학교 자연과학대학 수학과
- M. Sagae, "Upper and lower bounds for the arithmetic-geometric-harmonic means of positive definite matrices", Linear and Multilinear Alg., 37, pp.279~282 (1994)
- T. Yamazaki, "An extension of Kantorovich inequality to n-operators via the geometric mean by Ando-Li-Mathias", Linear Algebra Appl., 416, pp.688~695 (2006)