The arithmetic-geometric mean and its generalizations for noncommuting linear operators
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 15 (1988) no. 2, p. 239-308
@article{ASNSP_1988_4_15_2_239_0,
     author = {Nussbaum, Roger D. and Cohen, Joel E.},
     title = {The arithmetic-geometric mean and its generalizations for noncommuting linear operators},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola normale superiore},
     volume = {Ser. 4, 15},
     number = {2},
     year = {1988},
     pages = {239-308},
     zbl = {0756.47018},
     mrnumber = {1007399},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_1988_4_15_2_239_0}
}
Nussbaum, Roger D.; Cohen, Joel E. The arithmetic-geometric mean and its generalizations for noncommuting linear operators. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 15 (1988) no. 2, pp. 239-308. http://www.numdam.org/item/ASNSP_1988_4_15_2_239_0/

[1] W.N. Andersonjr. - R.J. Duffin, Series and parallel addition of matrices, J. Math. Anal. Appl. 26 (1969), pp. 576-594. | MR 242573 | Zbl 0177.04904

[2] W.N. Andersonjr. - G.E. Trapp, Shorted operators II, SIAM J. Appl. Math. 28 (1975), pp. 60-71. | MR 356949 | Zbl 0295.47032

[3] Tsuyoshi Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra and its Applications 27 (1979), pp. 203-241. | MR 535686 | Zbl 0495.15018

[4] -, On the arithmetic-geometric-harmonic mean inequality for positive definite matrices, Linear Algebra and its Applications 52-53 (1983), pp. 31-37. | MR 709342 | Zbl 0516.15011

[5] Tsuyoshi Ando - Fumio Kubo, Means of positive linear operators, Math. Ann. 246 (1980), pp. 205-244. | MR 563399 | Zbl 0412.47013

[6] J. Arazy - T. Claesson - S. Janson - J. Peetre, Means and their iterations, in Proceedings of the Nineteenth Nordic Congress of Mathematicians, Reykjavik, 1984, pp. 191-212 (published by the Icelandic Math. Soc., 1985). | MR 828035 | Zbl 0606.26007

[7] R. Arens, The analytic-functional calculus in commutative topological algebras, Pac. J. Math. 11 (1961), pp. 405-429. | MR 140960 | Zbl 0109.34203

[8] J. Bendat - S. Sherman, Monotone and convex operator functions, Trans. Amer. Math. Soc. 79 (1955), pp. 58-71. | MR 82655 | Zbl 0064.36901

[9] G. Borchardt, Gesammelte Werke, Herausgegeben von G. Hettner, Reimer. Berlin, 1888. | JFM 20.0015.01

[10] J.M. Borwein - P.B. Borwein, The arithmetic-geometric means and fast computation of elementary functions, SIAM Review 26 (1984), pp. 351-366. | MR 750454 | Zbl 0557.65009

[11] D. Borwein - P.B. Borwein, A generalized arithmetic-geometric mean, Problem 83-12, SIAM Review 25 (1983), pp. 201.

[12] B.C. Carlson, Hidden symmetries of special functions, SIAM Review 12 (1970), pp. 332-345. | MR 262554 | Zbl 0204.38503

[13] -, Algorithms involving arithmetic and geometric means, Amer. Math. Monthly 78 (1971), pp. 496-504. | MR 283246 | Zbl 0218.65035

[14] J.E. Cohen - R.D. Nussbaum, Arithmetic-geometric means of positive matrices, Math. Proc. Cambridge Phil. Soc. 101 (1987), pp. 209-219. | MR 870592 | Zbl 0623.15012

[15] D.A. Cox, The arithmetic-geometric mean of Gauss, Enseignement Math. 30 (1984), Pl275-330. | MR 767905 | Zbl 0583.33002

[16] W. Donoghue, Monotone Matrix Functions and Analytic Continuation, Springer-Verlag, New York, 1974. | MR 486556 | Zbl 0278.30004

[17] N. Dunford - J. Schwartz, Linear Operators, vol. 1, Interscience, New York, 1958. | Zbl 0084.10402

[18] C.J. Everett - N. Metropolis, A generalization of the Gauss limit for iterated means, Advances in Math. 7 (1971), pp. 297-300. | MR 296232 | Zbl 0221.40001

[19] Jun-Ichi Fujii, Arithmetic-geometric means of operators, Math. Japonica 23 (1979), pp. 667-669. | MR 529901 | Zbl 0407.47011

[20] G.H. Hardy - J.E. Littlewood - G. Polya, Inequalities, 2nd ed., Cambridge University Press, 1959. | Zbl 0047.05302

[21] T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, New York, 1984. | Zbl 0531.47014

[22] D.H. Lehmer, On the compounding of certain means, J. Math. Anal. Appl. 36 (1971), pp. 183-200. | MR 281696 | Zbl 0222.26018

[23] K. Lowner - C. Loewner, Uber monotone Matrixfunktionen, Math. Zeit. 38 (1934), pp. 177-216. | JFM 60.0055.01 | MR 1545446 | Zbl 0008.11301

[24] C. Loewner Some classes of functions defined by difference or differential inequalities, Bull. Amer. Math. Soc. 56 (1950), pp. 308-319. | MR 38387 | Zbl 0041.18202

[25] A.W. Marshall - I. Olkin, Inequalities: Theory of Majorization and its Applications, Academic Press, New York, 1979. | MR 552278 | Zbl 0437.26007

[26] R.D. Nussbaum, Iterated nonlinear maps and Hilbert's projective metric, Memoirs Amer. Math. Soc., no. 391 (1988). | Zbl 0666.47028

[27] -, Iterated nonlinear maps and Hilbert's projective metric: a summary, in Dynamics of Infinite Dimensional Systems, edited by S.N. Chow and J.K. Hale, Springer Verlag, Heidelberg, 1987, pp. 231-249.

[28] -, Convexity and log convexity for the spectral radius, Linear Alg. and Appl. 73 (1986), pp. 59-122. | MR 818894 | Zbl 0588.15016

[29] I. Olkin - J.W. Pratt, A multivariable Tchebycheff inequality, Annals of Math. Stat. 29 (1958), pp. 226-234. | MR 93865 | Zbl 0085.35204

[30] J. Peetre, Iteration of power means and other related means, preprint, University of Lund, Sweden, 1984.

[31] W. Pusz - S.L. Woronowicz, Functional calculus for sesquilinear forms, and the purification map, Rep. Math. Phys. 8 (1975), pp. 159-170. | MR 420302 | Zbl 0327.46032

[32] E.U. Stickel, Fast computation of matrix exponential and logarithm, Analysis (München) 5 (1985), pp. 163-173. | MR 791497 | Zbl 0543.65021

[33] A.E. Taylor, Introduction to Functional Analysis, Wiley, New York, 1958. | MR 98966 | Zbl 0081.10202

[34] P. Whittle, A multivariate generalization of Tchebychev's inequality, Quarterly J. of Math. Oxford, Series 2, 9 (1958), pp. 232-240. | MR 100321 | Zbl 0085.35205

[35] J. Wimp, Computation with Recurrence Relations, Pittman Advanced Publishing Program, Boston, 1984. | MR 727118 | Zbl 0543.65084

[36] K. Yoshida, Functional Analysis, 6th ed., Springer-Verlag, New York, 1980.