Embedding of a Uniquely Divisible Abelian Semigroup In a Convex Cone

I. V. Orlov

Matematičeskie zametki, Tome 102 (2017) no. 3, pp. 396-404.

Résumé

It is proved that every uniquely divisible Abelian semigroup admits an injective subadditive embedding in a convex cone. As an application, the classical theory of generators of one-parameter operator semigroups is generalized to the case in which the parameter ranges over a uniquely divisible semigroup.

Détail

Keywords: Abelian semigroup, unique divisibility, convex cone, infinitesimal generator of an operator semigroup.

Bibliographie

[1] G. Birkhoff, “Integration of functions with values in a Banach space”, Trans. Amer. Math. Soc., 38:2 (1935), 357–378 | MR | Zbl

[2] J. H. Rådström, “An embedding theorem for space of convex sets”, Proc. Amer. Math. Soc., 3 (1952), 165–169 | DOI | MR | Zbl

[3] G. Godini, “A framework for best simultaneous approximation, normed almost linear spaces”, J. Approx. Theory, 43:4 (1985), 338–358 | DOI | MR | Zbl

[4] R. E. Worth, “Boundaries of semilinear spaces and semialgebras”, Trans. Amer. Math. Soc., 148 (1970), 99–119 | DOI | MR | Zbl

[5] R. Urbański, “A generalization of the Minkovski–Rädström–Hörmander theorem”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 24:9 (1976), 709–715 | MR | Zbl

[6] K. Keimel, W. Roth, Ordered Cones and Approximation, Lecture Notes in Math., 1517, Springer-Verlag, Berlin, 1992 | DOI | MR | Zbl

[7] B. Fuchssteiner, W. Lusky, Convex Cones, North Holland Math. Stud., 56, North Holland, Amsterdam, 1981 | MR | Zbl

[8] T. Abreu, E. Corbacho, V. Tarieladze, “Uniform type conoids”, Math. Pannon., 19:2 (2008), 155–170 | MR

[9] E. Corbacho Rosas, D. Dikranjan, V. Tarieladze, “Absorption adjunctable semigroups”, Nuclear Groups and Lie Groups, Res. Exp. Math., 24, Heldermann Verlag, Lemgo, 2001, 77–103 | MR | Zbl

[10] I. V. Orlov, “Vvedenie v sublineinyi analiz”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 53, RUDN, M., 2014, 64–132 | MR | Zbl

[11] I. V. Orlov, “Teoremy ob obratnoi i neyavnoi funktsiyakh v klasse subgladkikh otobrazhenii”, Matem. zametki, 99:4 (2016), 631–634 | DOI | MR | Zbl

[12] F. S. Stonyakin, “An analogue of the Hahn–Banach theorem for functionals on abstract convex cones”, Eurasian Math. J., 7:3 (2016), 89–99 | MR

[13] M. L. Goldman, P. P. Zabreiko, “Optimalnoe banakhovo funktsionalnoe prostranstvo dlya konusa neotritsatelnykh ubyvayuschikh funktsii”, Tr. In-ta matem., 22:1 (2014), 24–34

[14] V. D. Stepanov, “Ob optimalnykh prostranstvakh Banakha, soderzhaschikh vesovoi konus monotonnykh ili kvazivognutykh funktsii”, Matem. zametki, 98:6 (2015), 907–922 | DOI | MR | Zbl

[15] E. G. Bakhtigareeva, M. L. Goldman, “Assotsiirovannye normy i optimalnye vlozheniya dlya odnogo klassa dvukhvesovykh integralnykh kvazinorm”, Fundament. i prikl. matem., 19:5 (2014), 3–33 | MR

[16] E. Bakhtigareeva, “Optimal Banach function space for a given cone of decreasing functions in a weighted $L_p$-space”, Eurasian Math. J., 6:1 (2015), 6–25 | MR

[17] A. Löhne, On Convex Functions with Values in Semi-Linear Spaces, , 2004 http://ito.mathematik.uni-halle.de

[18] A. Hamel, A. Löhne, Minimal Set Theorems, http://ito.mathematik.uni-halle.de/~loehne/pdf/mst_web.pdf

[19] H. Cartan, S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton, NJ, 1999 | MR | Zbl

[20] S. Feigelstock, “Divisible is injective”, Soochow J. Math., 32:2 (2006), 241–243 | MR | Zbl

[21] C. Casacuberta, J. L. Rodríguez, L. J. Hernándes Paricio, “Models for torsion homotopy types”, Israel J. Math., 107 (1998), 301–318 | DOI | MR | Zbl

[22] B. V. Gnedenko, Kurs teorii veroyatnostei, Nauka, M., 1965 | MR | Zbl

[23] S. M. Ross, Introduction to Probability Models, Academic Press, Boston, MA, 1993 | MR | Zbl

[24] E. Khille, R. Fillips, Funktsionalnyi analiz i polugruppy, IL, M., 1962 | MR | Zbl

[25] R. V. Shamin, Polugruppy operatorov, RUDN, M., 2008