On some Banach lattice-valued operators: A Survey

Yıl 2017, Cilt: 1 Sayı: 1, 14 - 40, 30.09.2017

Nutefe Kwami Agbeko

https://doi.org/10.31197/atnaa.338349

Öz

In 1928, at the International Mathematical Congress held in Bologna (Italy), Frigyes Riesz introduced

the notion of vector lattice on function spaces and, talked about linear operators that preserve the join

operation, nowadays known in the literature as Riesz homomorphisms (see [32]). In this survey we review

the behaviors of some non-linear join-preserving Riesz space-valued functions, and we show how existing

addition dependent results can be proved in these environments mutatis mutandis. (We kindly refer the

reader to the papers [1, 2, 3, 4, 6, 7, 8, 9, 10, 5] for more information.)

Anahtar Kelimeler

Banach lattice-valued operators; , anach lattices , optimal measure , optimal average , dual Orlicz spaces , functional equation , functional inequality , Hyers-Ulam-Aoki type of stability;

Kaynakça

• N. K. Agbeko, On optimal averages, Acta Math. Hung. 63 (1-2)(1994), 1-15.
• N. K. Agbeko, On the structure of optimal measures and some of its applications, Publ. Math. Debrecen 46/1-2 (1995), 79-87.
• N. K. Agbeko, How to characterize some properties of measurable functions, Math. Notes, Miskolc 1/2 (2000), 87-98.
• N. K. Agbeko, Mapping bijctively -algebras onto power sets, Math. Notes, Miskolc 2/2 (2001), 85-92.
• N. K. Agbeko and A. Hazy, An algorithmic determination of optimal measure form data and some applications, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis 26(2010), 99-111. ISSN 1786-0091
• Agbeko, Nutefe Kwami, Stability of maximum preserving functional equations on banach lattices, Miskolc Math. Notes, 13(2012), No. 2, 187-196.
• Nutefe Kwami Agbeko, Sever Silvestru Dragomir, The extension of some Orlicz space results to the theory of optimal measure, Math. Nachr. 286(2013), No 8-9, 760-771 / DOI 10.1002/mana.201200066.
• N. K. Agbeko, The Hyers-Ulam-Aoki type stability of some functional equation on Banach lattices, Bull. Polish Acad. Sci. Math., 63, No. 2, (2015), 177-184. DOI: 10.4064/ba63-2-6
• N.K. Agbeko, A remark on a result of Schwaiger, Indag. Math. 28 , Issue 2, (2017), 268-275. [http://dx.doi.org/10.1016/j.indag.2016.06.013]
• N.K. Agbeko, W. Fechner, and E. Rak, On lattice-valued maps stemming from the notion of optimal average, Acta Math. Hungar, 152(2017), No 1, 72-83.
• C.D. Aliprantis and O. Burkinshaw, Locally solid Riesz spaces with applications to economics, Mathematical Surveys and Monographs, vol. 105, 2nd Edition, American Mathematical Society, Providence, RI, 2003. ISBN 0-8218-3408-8.
• T. Aoki, Stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2(1950), no. 1-2, 64-66.
• D. H. Hyers, On the Stability of the Linear Functional Equation, Proceedings of the National Academy of Sciences 27(1941). DOI 10.2307/87271.
• Csaszar, A. and Laczkovich, M.: Discrete and equal convergence, Studia Sci. Math. Hungar., 10(1975), 463-472.
• Csaszar, A. and Laczkovich, M.: Some remarks on discrete Baire classes, Acta Math. Acad. Sci. Hungar., 33(1979), 51-70.
• Csaszar, A. and Laczkovich, M.: Discrete and equal Baire classes, Acta Math. Hung.,55(1990), 165-178.
• I. Fazekas, A note on "optimal measures", Publ. Math. Debrecen 51 / 3-4(1997), 273-277.
• Gian-Luigi Forti, Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations, J. Math. Anal. Appl. 295(2004), 127-133.
• D.H. Fremlin, Topological Riesz spaces and measure theory, Cambridge University Press, London-New York, 1974.
• Hans Freudenthal Topologische Gruppen mit gengend vielen fastperiodischen Funktionen. (German) Ann. of Math. (2)37(1936), no. 1, 57-77.
• Z. Gajda, On stability of additive mappings, Internat. J. Math. Sci., 14(1991),No. 3, 431{434.
• Roman Ger and Peter Semrl, The stability of the exponential equation, Proc. Amer. Math. Soc. 124(1996), no. 3, 779-787.
• Hyers, D. H., On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27(1941), 222-224.
• L.V. Kantorovich, Sur les proprietes des espaces semi-ordonnes lineaires, C.R. Acad. Sci. Paris Ser. A-B 202(1936), 813-816.
• L.V. Kantorovich, Concerning the general theory of operations in partially ordered spaces, DAN SSSR 1(1936), 271-274. (In Russian).
• M. A. Krasnosel'ski and B. Ya Rutickii, Convex functions and Orlicz-spaces. (Transl. from Russian by Boron L. F.) Noordho, Groningen, 1961.
• M. Laczkovich, The local stability of convexity, anity and of the Jensen equation, Aequationes Math. 58(1999), 135-142.
• W.A.J. Luxemburg and A.C. Zaanen, Riesz spaces, Vol I, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1971.
• Gy. Maksa, The stability of the entropy of degree alpha, J. Math. Anal. Appl. 346(2008), no. 1, 17-21.
• J. Neveu, Martingales a temps discret, Masson et Cie, 1972.
• Zs. Pales, Hyers-Ulam stability of the Cauchy functional equation on square-symmetric groupoids, Publ. Math. Debrecen 58(2001), 651-666.
• Frigyes Riesz, Sur la decomposition des operations fonctionnelles lineaires, Atti. Congr. Internaz. Mat. Bologna, 3 (1930), 143-148. [http://www.mathunion.org/ICM/ICM1928.3/Main/icm1928.3.0143.0148.ocr.pdf]
• Henrik Kragh Sorensen, Exceptions and counterexamples: Understanding Abel's comment on Cauchy'Theorem, Historica Mathematica 32(2005), 453-480.
• J. Schwaiger, Remark 10, Report of 30th Internat. Symp. on Functional Equations, Aeq. Math. 46(1993), 289.
• Laszlo Szekelyhidi, Stability of functional equations on hypergroups, Aequationes Mathematicae, (2015), 1-9.
• S. M. Ulam, A Collection of the Mathematical Problems, Interscience, New York, 1960.