×

Normal cones and Thompson metric. (English) Zbl 1330.46007

Rassias, Themistocles M. (ed.) et al., Topics in mathematical analysis and applications. Cham: Springer (ISBN 978-3-319-06553-3/hbk; 978-3-319-06554-0/ebook). Springer Optimization and Its Applications 94, 209-258 (2014).
Summary: The aim of this paper is to study the basic properties of the Thompson metric \(d_T\) in the general case of a linear space \(X\) ordered by a cone \(K\). We show that \(d_T\) has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of \(d_T\), and some results concerning the topology of \(d_T\), including a brief study of the \(d_T\)-convergence of monotone sequences. It is shown that most results are true without any assumption of an Archimedean-type property for \(K\). One considers various completeness properties and one studies the relations between them. Since \(d_T\) is defined in the context of a generic ordered linear space, with no need of an underlying topological structure, one expects to express its completeness in terms of properties of the ordering with respect to the linear structure. This is done in this paper and, to the best of our knowledge, this has not been done yet. The Thompson metric \(d_T\) and order-unit (semi)norms \(|\cdot|_u\) are strongly related and share important properties, as both are defined in terms of the ordered linear structure. Although \(d_T\) and \(|\cdot|_u\) are only topologically (and not metrically) equivalent on \(K_u\), we prove that the completeness is a common feature. One proves the completeness of the Thompson metric on a sequentially complete normal cone in a locally convex space. At the end of the paper, it is shown that, in the case of a Banach space, the normality of the cone is also necessary for the completeness of the Thompson metric.
For the entire collection see [Zbl 1301.00059].

MSC:

46A40 Ordered topological linear spaces, vector lattices
46A03 General theory of locally convex spaces
46B40 Ordered normed spaces
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Akian, M., Gaubert, S., Nussbaum, R.: Uniqueness of the fixed point of nonexpansive semidifferentiable maps (2013). arXiv:1201.1536v2 · Zbl 1357.47056
[2] Aliprantis, C.D., Tourky, R.: Cones and Duality. Graduate Studies in Mathematics, vol. 84. American Mathematical Society, Providence (2007) · Zbl 1127.46002
[3] Bauer, H.; Bear, H. S., The part metric in convex sets, Pac. J. Math., 30, 15-33 (1969) · Zbl 0176.42801 · doi:10.2140/pjm.1969.30.15
[4] Bear, H. S.; Weiss, M. L., An intrinsic metric for parts, Proc. Am. Math. Soc., 18, 812-817 (1967) · Zbl 0184.34303 · doi:10.1090/S0002-9939-1967-0215043-1
[5] Birkhoff, G., Extensions of Jentzsch’s theorem, Trans. Am. Math. Soc., 85, 219-227 (1957) · Zbl 0079.13502
[6] Breckner, W. W., Rational s-Convexity: A Generalized Jensen-Convexity (2011), Cluj-Napoca: Cluj University Press, Cluj-Napoca · Zbl 1245.46001
[7] Bushell, P. J., Hilbert’s projective metric and positive contraction mappings in a Banach space, Arch. Ration. Mech. Anal., 52, 330-338 (1973) · Zbl 0275.46006 · doi:10.1007/BF00247467
[8] Bushell, P. J., On the projective contraction ratio for positive linear mappings, J. Lond. Math. Soc., 6, 2, 256-258 (1973) · Zbl 0255.47048 · doi:10.1112/jlms/s2-6.2.256
[9] Chen, Y.-Z., Thompson’s metric and mixed monotone operators, J. Math. Anal. Appl., 177, 31-37 (1993) · Zbl 0804.47054 · doi:10.1006/jmaa.1993.1241
[10] Chen, Y.-Z., A variant of the Meir-Keeler-type theorem in ordered Banach spaces, J. Math. Anal. Appl., 236, 585-593 (1999) · Zbl 0952.47042 · doi:10.1006/jmaa.1999.6469
[11] Chen, Y.-Z., On the stability of positive fixed points, Nonlinear Anal. Theory Methods Appl., 47, 2857-2862 (2001) · Zbl 1042.47513 · doi:10.1016/S0362-546X(01)00405-9
[12] Chen, Y.-Z., Stability of positive fixed points of nonlinear operators, Positivity, 6, 47-57 (2002) · Zbl 1008.47048 · doi:10.1023/A:1012079817987
[13] Deimling, K., Nonlinear Functional Analysis (1985), Berlin: Springer, Berlin · Zbl 0559.47040 · doi:10.1007/978-3-662-00547-7
[14] Guo, D.; Cho, Y. J.; Zhu, J., Partial Ordering Methods in Nonlinear Problems (2004), Hauppauge: Nova Science Publishers, Hauppauge · Zbl 1116.45007
[15] Hatori, O.; Molnár, L., Isometries of the unitary groups and Thompson isometries of the spaces of invertible positive elements in C^∗-algebras, J. Math. Anal. Appl., 409, 158-167 (2014) · Zbl 1306.46053 · doi:10.1016/j.jmaa.2013.06.065
[16] Hilbert, D., Über die gerade Linie als kürzeste Verbindung zweier Punkte, Math. Ann., 46, 91-96 (1895) · JFM 26.0540.02 · doi:10.1007/BF02096204
[17] Hyers, D. H.; Isac, G.; Rassias, T. M., Topics in Nonlinear Analysis & Applications (1997), River Edge: World Scientific, River Edge · Zbl 0878.47040 · doi:10.1142/2998
[18] Izumino, S.; Nakamura, N., Geometric means of positive operators, II. Sci. Math. Jpn., 69, 35-44 (2009) · Zbl 1182.47020
[19] Jameson, G.: Ordered Linear Spaces. Lecture Notes in Mathematics, vol. 141. Springer, Berlin (1970) · Zbl 0196.13401
[20] Jung, C. F.K., On generalized complete metric spaces, Bull. Am. Math. Soc., 75, 113-116 (1969) · Zbl 0194.23801 · doi:10.1090/S0002-9904-1969-12165-8
[21] Köthe, G., Topological Vector Spaces I (1969), Berlin: Springer, Berlin · Zbl 0179.17001
[22] Krause, U.; Nussbaum, R. D., A limit set trichotomy for self-mappings of normal cones in Banach spaces, Nonlinear Anal. Theory Methods Appl., 20, 855-870 (1993) · Zbl 0833.47047 · doi:10.1016/0362-546X(93)90074-3
[23] Lemmens, B., Nussbaum, R.D.: Nonlinear Perron-Frobenius Theory. Cambridge Tracts in Mathematics, vol. 189. Cambridge University Press, Cambridge (2012) · Zbl 1246.47001
[24] Lemmens, B., Nussbaum, R.D.: Birkhoff’s version of Hilbert’s metric and its applications in analysis (2013). arXiv:1304.7921 · Zbl 1282.47068
[25] Lins, B., A Denjoy-Wolff theorem for Hilbert metric nonexpansive maps on polyhedral domains, Math. Proc. Camb. Philos. Soc., 143, 157-164 (2007) · Zbl 1127.47048 · doi:10.1017/S0305004107000199
[26] Lins, B.; Nussbaum, R. D., Iterated linear maps on a cone and Denjoy-Wolff theorems, Linear Algebra Appl., 416, 615-626 (2006) · Zbl 1110.15016 · doi:10.1016/j.laa.2005.12.009
[27] Lins, B.; Nussbaum, R. D., Denjoy-Wolff theorems, Hilbert metric nonexpansive maps and reproduction-decimation operators. J. Funct. Anal., 254, 2365-2386 (2008) · Zbl 1143.47036
[28] Molnár, L., Thompson isometries of the space of invertible positive operators, Proc. Am. Math. Soc., 137, 11, 3849-3859 (2009) · Zbl 1184.46021 · doi:10.1090/S0002-9939-09-09963-8
[29] Nakamura, N., Geometric means of positive operators, Kyungpook Math. J., 49, 167-181 (2009) · Zbl 1182.47021 · doi:10.5666/KMJ.2009.49.1.167
[30] Ng, K. F., On order and topological completeness, Math. Ann., 196, 171-176 (1972) · Zbl 0221.46010 · doi:10.1007/BF01419613
[31] Nussbaum, R. D., Hilbert’s projective metric and iterated nonlinear maps, Mem. Am. Math. Soc., 75, 391, 4-137 (1988) · Zbl 0666.47028
[32] Nussbaum, R. D., Iterated nonlinear maps and Hilbert’s projective metric, Mem. Am. Math. Soc., 79, 401, 4-118 (1989) · Zbl 0669.47031
[33] Nussbaum, R. D., Fixed point theorems and Denjoy-Wolff theorems for Hilbert’s projective metric in infinite dimensions, Topol. Methods Nonlinear Anal., 29, 199-249 (2007) · Zbl 1143.47037
[34] Nussbaum, R.D., Walsh, C.A.: A metric inequality for the Thompson and Hilbert geometries. JIPAM J. Inequal. Pure Appl. Math. 5, 14 pp. (2004). Article 54 · Zbl 1061.53051
[35] Peressini, A. L., Ordered Topological Vector Spaces (1967), New York: Harper & Row, New York · Zbl 0169.14801
[36] Pérez Carreras, P., Bonet, J.: Barrelled Locally Convex Spaces. North-Holland Mathematics Studies, vol. 131 (Notas de Matemática, vol. 113). North-Holland, Amsterdam (1987) · Zbl 0614.46001
[37] Rus, M.-D.: The method of monotone iterations for mixed monotone operators. Babeş-Bolyai University, Ph.D. thesis, Cluj-Napoca (2010)
[38] Rus, M.-D., Fixed point theorems for generalized contractions in partially ordered metric spaces with semi-monotone metric, Nonlinear Anal. Theory Methods Appl., 74, 1804-1813 (2011) · Zbl 1221.54072 · doi:10.1016/j.na.2010.10.053
[39] Samelson, H., On the Perron-Frobenius theorem, Mich. Math. J., 4, 57-59 (1957) · Zbl 0077.02303 · doi:10.1307/mmj/1028990177
[40] Schaefer, H.H.: Topological Vector Spaces. Third printing corrected. Graduate Texts in Mathematics, vol. 3. Springer, New York (1971) · Zbl 0217.16002
[41] Thompson, A. C., On certain contraction mappings in a partially ordered vector space, Proc. Am. Math. Soc., 14, 438-443 (1963) · Zbl 0147.34903
[42] Turinici, M., Maximal elements in a class of order complete metric spaces, Math. Jpn., 25, 511-517 (1980) · Zbl 0452.54025
[43] Wong, Y. C., Relationship between order completeness and topological completeness, Math. Ann., 199, 73-82 (1972) · Zbl 0233.46013 · doi:10.1007/BF01419577
[44] Wong, Y. C.; Ng, K. F., Partially Ordered Topological Vector Spaces (1973), Oxford Mathematical Monographs: Clarendon Press, Oxford, Oxford Mathematical Monographs · Zbl 0269.46007
[45] Zălinescu, C., Convex Analysis in General Vector Spaces (2002), River Edge: World Scientific, River Edge · Zbl 1023.46003 · doi:10.1142/9789812777096
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.