Geometric Means and Tracy-Singh Products for Positive Operators

Arnon Ploymukda, Pattrawut Chansangiam

Abstract


We investigate relationship between metric/spectral/Sagae-Tanabe geometric means for several positive operators and Tracy-Singh products in terms of identities and inequalities. In particular, we obtain various generalizations of arithmetic-geometric-harmonic means inequality and its reverse. Moreover, we introduce the weighted Sagae-Tanabe spectral geometric mean for several positive operators and deduce its properties related to Tracy-Singh products.

Keywords


Metric (spectral) geometric mean; Sagae-Tanabe metric (spectral) geometric mean; Tensor product; Tracy-Singh product; Khatri-Rao product

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References


E. Ahn, S. Kim, H. Lee and Y. Lim, Sagae-Tanabe weighted mean and reverse inequalities, Kyungpook Math. J. 47 (2007), 595 – 600.

M. Alic, B. Mond, J. Pecaric and V. Volenec, The arithmetic-geometric-harmonic-mean and related matrix inequalities, Linear Algebra and its Applications 40264 (1997), 55 – 62.

T. Ando, Topics on Operator Inequalities, Lecture Notes Hokkaido Univ., Sapporo (1978).

T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl. 26 (1979), 203 – 241.

R. Bhatia, Positive Definite Matrices, Princeton University Press, New Jersey (2007).

M. Fiedler and V. Pták, A new positive definite geometric mean of two positive definite matrices, Linear Algebra Appl. 251 (1997), 1 – 20.

J.I. Fujii, Y. Seo and M. Tominaga, Kantorovich type operator inequalities via the Specht ratio, Linear Algebra and its Applications 377 (2004), 69 – 81.

A. Kilicman and Z. Al-Zhour, Improvements on geometric means related to the Tracy-Singh products of positive matrices, Mathematica 21 (2005), 49–65.

S. Lang, Fundamentals of Differential Geometry, Graduate Texts in Mathematics, Springer, Heidelberg (1999).

J. Lawson and Y. Lim, The geometric mean, matrices, metrics, and more, Am. Math. Monthly 108 (2001), 797 – 812.

A. Ploymukda and P. Chansangiam, Khatri-Rao products of operator matrices acting on the direct sum of Hilbert spaces, Journal of Mathematics 7 pages (2016), DOI: 10.1155/2016/8301709.

A. Ploymukda, W. Lewkeeratiyutkul and P. Chansangaim, Algebraic and order properties of Tracy-Singh product for operator matrices, J. Comput. Anal. Appl. 24(4) (2018), 656 – 664.

A. Ploymukda, W. Lewkeeratiyutkul and P. Chansangaim, Analytic properties of Tracy-Singh product for operator matrices, J. Comput. Anal. Appl. 24(4) (2018), 665 – 674.

A. Ploymukda and P. Chansangaim, Khatri-Rao products and selection operators, J. Comput. Anal. Appl. 27(2) (2019), 316 – 325.

M. Sagae and K. Tanabe, Upper and lower bounds for the arithmetic-geometric-harmonic means of positive definite matrices, Linear and Multilinear Algebra 37 (1994), 279 – 282.

A.C. Thompson, On certain contraction mappings in a partially ordered vector space, Proc. Amer. Math. Soc. 14 (1963), 438 – 443.

D.S. Tracy and R.P. Singh, A new matrix product and its applications in partitioned matrix differentiation, Stat. Neerl. 26 (1972), 143 – 157.

J. Zanni and C.S. Kubrusly, A note on compactness of tensor products, Acta Math. Univ. Comenianae 84 (2015), 59 – 62.


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