Karcher means and Karcher equations of positive definite operators

Lawson, Jimmie; Lim, Yongdo

doi:10.1090/S2330-0000-2014-00003-4

Karcher means and Karcher equations of positive definite operators
HTML articles powered by AMS MathViewer

by Jimmie Lawson and Yongdo Lim HTML | PDF

Trans. Amer. Math. Soc. Ser. B 1 (2014), 1-22

Abstract:

The Karcher or least-squares mean has recently become an important tool for the averaging and studying of positive definite matrices. In this paper we show that this mean extends, in its general weighted form, to the infinite-dimensional setting of positive operators on a Hilbert space and retains most of its attractive properties. The primary extension is via its characterization as the unique solution of the corresponding Karcher equation. We also introduce power means $P_t$ in the infinite-dimensional setting and show that the Karcher mean is the strong limit of the monotonically decreasing family of power means as $t\to 0^+$. We show that each of these characterizations provide important insights about the Karcher mean.

References

T. Ando, Chi-Kwong Li, and Roy Mathias, Geometric means, Linear Algebra Appl. 385 (2004), 305–334. MR 2063358, DOI 10.1016/j.laa.2003.11.019
Marcel Berger, A panoramic view of Riemannian geometry, Springer-Verlag, Berlin, 2003. MR 2002701, DOI 10.1007/978-3-642-18245-7
Rajendra Bhatia, On the exponential metric increasing property, Linear Algebra Appl. 375 (2003), 211–220. MR 2013466, DOI 10.1016/S0024-3795(03)00647-5
Rajendra Bhatia, Positive definite matrices, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2007. MR 2284176
Rajendra Bhatia and John Holbrook, Riemannian geometry and matrix geometric means, Linear Algebra Appl. 413 (2006), no. 2-3, 594–618. MR 2198952, DOI 10.1016/j.laa.2005.08.025
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
G. Corach, H. Porta, and L. Recht, Convexity of the geodesic distance on spaces of positive operators, Illinois J. Math. 38 (1994), no. 1, 87–94. MR 1245836, DOI 10.1215/ijm/1255986889
Karl E. Gustafson and Duggirala K. M. Rao, Numerical range, Universitext, Springer-Verlag, New York, 1997. The field of values of linear operators and matrices. MR 1417493, DOI 10.1007/978-1-4613-8498-4
Richard V. Kadison, Strong continuity of operator functions, Pacific J. Math. 26 (1968), 121–129. MR 231211, DOI 10.2140/pjm.1968.26.121
H. Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math. 30 (1977), no. 5, 509–541. MR 442975, DOI 10.1002/cpa.3160300502
Fumio Kubo and Tsuyoshi Ando, Means of positive linear operators, Math. Ann. 246 (1979/80), no. 3, 205–224. MR 563399, DOI 10.1007/BF01371042
Serge Lang, Fundamentals of differential geometry, Graduate Texts in Mathematics, vol. 191, Springer-Verlag, New York, 1999. MR 1666820, DOI 10.1007/978-1-4612-0541-8
Jimmie D. Lawson and Yongdo Lim, The geometric mean, matrices, metrics, and more, Amer. Math. Monthly 108 (2001), no. 9, 797–812. MR 1864051, DOI 10.2307/2695553
Jimmie Lawson and Yongdo Lim, Metric convexity of symmetric cones, Osaka J. Math. 44 (2007), no. 4, 795–816. MR 2383810
Jimmie Lawson and Yongdo Lim, A general framework for extending means to higher orders, Colloq. Math. 113 (2008), no. 2, 191–221. MR 2425082, DOI 10.4064/cm113-2-3
Jimmie Lawson and Yongdo Lim, Monotonic properties of the least squares mean, Math. Ann. 351 (2011), no. 2, 267–279. MR 2836658, DOI 10.1007/s00208-010-0603-6
Yongdo Lim and Miklós Pálfia, Matrix power means and the Karcher mean, J. Funct. Anal. 262 (2012), no. 4, 1498–1514. MR 2873848, DOI 10.1016/j.jfa.2011.11.012
Maher Moakher, A differential geometric approach to the geometric mean of symmetric positive-definite matrices, SIAM J. Matrix Anal. Appl. 26 (2005), no. 3, 735–747. MR 2137480, DOI 10.1137/S0895479803436937
Roger D. Nussbaum and Joel E. Cohen, The arithmetic-geometric mean and its generalizations for noncommuting linear operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), no. 2, 239–308 (1989). MR 1007399
Roger D. Nussbaum, Finsler structures for the part metric and Hilbert’s projective metric and applications to ordinary differential equations, Differential Integral Equations 7 (1994), no. 5-6, 1649–1707. MR 1269677
Karl-Theodor Sturm, Probability measures on metric spaces of nonpositive curvature, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002) Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 357–390. MR 2039961, DOI 10.1090/conm/338/06080
A. C. Thompson, On certain contraction mappings in a partially ordered vector space. , Proc. Amer. Math. soc. 14 (1963), 438–443. MR 0149237, DOI 10.1090/S0002-9939-1963-0149237-7
Joachim Weidmann, Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Springer-Verlag, New York-Berlin, 1980. Translated from the German by Joseph Szücs. MR 566954, DOI 10.1007/978-1-4612-6027-1
T. Yamazaki, The Riemannian mean and matrix inequalities related to the Ando-Hiai inequality and chaotic order, Operators and Matrices 6 (2012), 577–588.