The Kantorovich metric provides a way of measuring the distance between two Borel probability measures on a metric space. This metric has a broad range of applications from bioinformatics to image processing, and is commonly linked to the optimal transport problem in computer science (Deng and Du in Electron. Notes Theor. Comput. Sci. 253: 73–82, 2009; Villani in Optimal Transport: Old and New, Grundlehren der mathematischen Wissenschaften, vol. 338, 2009). Noteworthy to this paper will be the role of the Kantorovich metric in the study of iterated function systems, which are families of contractive mappings on a complete metric space. When the underlying metric space is compact, it is well known that the space of Borel probability measures on this metric space, equipped with the Kantorovich metric, constitutes a compact, and thus complete metric space. In previous work, we generalized the Kantorovich metric to operator-valued measures for a compact underlying metric space, and applied this generalized metric to the setting of iterated function systems (Davison in Acta Appl. Math., 2014, https://doi.org/10.1007/s10440-014-9976-y; Generalizing the Kantorovich Metric to Projection-Valued Measures: With an Application to Iterated Function Systems, 2015; Acta Appl. Math., 2018, https://doi.org/10.1007/s10440-018-0161-6). We note that the work of P. Jorgensen, K. Shuman, and K. Kornelson provided the framework for our application to this setting (Jorgensen in Adv. Appl. Math. 34(3):561–590, 2005; Jorgenson et al. in J. Math. Phys. 48(8):083511, 2007; Jorgensen in Operator Theory, Operator Algebras, and Applications, Contemp. Math., vol. 414, pp. 13–26, 2006). The situation when the underlying metric space is complete, but not necessarily compact, has been studied by A. Kravchenko (Sib. Math. J. 47(1), 68–76, 2006). In this paper, we extend the results of Kravchenko to the generalized Kantorovich metric on operator-valued measures.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Ali, S.: A geometrical property of POV measures, and systems of covariance. In: Differential Geometric Methods in Mathematical Physics. Lecture Notes in Mathematics, vol. 905, pp. 207–228 (1982)
Berberian, S.: Notes on Spectral Theory, 2nd edn. (2009)
Bogachev, V.: Measure Theory: Volume II. Springer, New York (2000)
Conway, J.: A Course in Functional Analysis, 2nd edn. Springer, New York (2000)
Davison, T.: Generalizing the Kantorovich metric to projection-valued measures. Acta Appl. Math. (2014). https://doi.org/10.1007/s10440-014-9976-y
Davison, T.: Erratum to: generalizing the Kantorovich metric to projection-valued measures. Acta Appl. Math. (2015). https://doi.org/10.1007/s10440-015-0018-1
Davison, T.: Generalizing the Kantorovich Metric to Projection-Valued Measures: With an Application to Iterated Function Systems. University of Colorado, Boulder (2015), ProQuest Dissertations Publishing
Davison, T.: A positive operator-valued measure for an iterated function system. Acta Appl. Math. (2018). https://doi.org/10.1007/s10440-018-0161-6
Deng, Y., Du, W.: The Kantorovich metric in computer science: a brief survey. Electron. Notes Theor. Comput. Sci. 253, 73–82 (2009)
Hutchinson, J.: Fractals and self similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)
Jorgensen, P.: Measures in wavelet decompositions. Adv. Appl. Math. 34(3), 561–590 (2005)
Jorgensen, P.: Use of operator algebras in the analysis of measures from wavelets and iterated function system. In: Operator Theory, Operator Algebras, and Applications. Contemp. Math., vol. 414, pp. 13–26. Am. Math. Soc., Providence (2006)
Jorgenson, P., Kornelson, K., Shuman, K.: Harmonic analysis of iterated function systems with overlap. J. Math. Phys. 48(8), 083511 (2007), 35
Kravchenko, A.S.: Completeness of the space of separable measures in the Kantorovich–Rubinshtein metric. Sib. Math. J. 47(1), 68–76 (2006)
Villani, C.: Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften, vol. 338. Springer, Berlin (2009)
Werner, R.F.: The uncertainty relation for joint measurement of position and momentum. J. Quantum Inf. Comput. 4(6), 546–562 (2004)
I would like to recognize my graduate advisor Judith Packer for her excellent guidance during my time in graduate school.
About this article
Cite this article
Davison, T. An Operator-Valued Kantorovich Metric on Complete Metric Spaces. Acta Appl Math 163, 49–72 (2019). https://doi.org/10.1007/s10440-018-0213-y
- Kantorovich metric
- Operator-valued measure
- Hutchinson measure
- Iterated Function System
- Fixed point
Mathematics Subject Classification