# On separation theorems for subadditive and superadditive functionals

Zbigniew Gajda; Zygfryd Kominek

Studia Mathematica (1991)

- Volume: 100, Issue: 1, page 25-38
- ISSN: 0039-3223

## Access Full Article

top## Abstract

top## How to cite

topGajda, Zbigniew, and Kominek, Zygfryd. "On separation theorems for subadditive and superadditive functionals." Studia Mathematica 100.1 (1991): 25-38. <http://eudml.org/doc/215871>.

@article{Gajda1991,

abstract = {We generalize the well known separation theorems for subadditive and superadditive functionals to some classes of not necessarily Abelian semigroups. We also consider the problem of supporting subadditive functionals by additive ones in the not necessarily commutative case. Our results are motivated by similar extensions of the Hyers stability theorem for the Cauchy functional equation. In this context the so-called weakly commutative and amenable semigroups appear naturally. The relations between these two classes of semigroups are discussed at the end of the paper.},

author = {Gajda, Zbigniew, Kominek, Zygfryd},

journal = {Studia Mathematica},

keywords = {separation; semigroup; subadditive functional; superadditive functional; Hyers-Ulam stability; Cauchy equation},

language = {eng},

number = {1},

pages = {25-38},

title = {On separation theorems for subadditive and superadditive functionals},

url = {http://eudml.org/doc/215871},

volume = {100},

year = {1991},

}

TY - JOUR

AU - Gajda, Zbigniew

AU - Kominek, Zygfryd

TI - On separation theorems for subadditive and superadditive functionals

JO - Studia Mathematica

PY - 1991

VL - 100

IS - 1

SP - 25

EP - 38

AB - We generalize the well known separation theorems for subadditive and superadditive functionals to some classes of not necessarily Abelian semigroups. We also consider the problem of supporting subadditive functionals by additive ones in the not necessarily commutative case. Our results are motivated by similar extensions of the Hyers stability theorem for the Cauchy functional equation. In this context the so-called weakly commutative and amenable semigroups appear naturally. The relations between these two classes of semigroups are discussed at the end of the paper.

LA - eng

KW - separation; semigroup; subadditive functional; superadditive functional; Hyers-Ulam stability; Cauchy equation

UR - http://eudml.org/doc/215871

ER -

## References

top- [1] A. Chaljub-Simon und P. Volkmann, Bemerkungen zu einem Satz von Rodé, manuscript. Zbl0712.39029
- [2] M. M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509-544. Zbl0078.29402
- [3] G. L. Forti, Remark 11, Report of Meeting, the 22nd Internat Symposium on Functional Equations, Aequationes Math. 29 (1985), 90-91.
- [4] G. L. Forti, The stability of homomorphisms and amenability with applications to functional equations, Università degli Studi di Milano, Quaderno 24, 1986.
- [5] Z. Gajda, Invariant means and representations of semigroups in the theory of functional equations, submitted.
- [6] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. I, Springer, Berlin 1963. Zbl0115.10603
- [7] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224. Zbl0061.26403
- [8] R. Kaufman, Interpolation of additive functionals, Studia Math. 27 (1966), 269-272. Zbl0143.36302
- [9] H. König, On the abstract Hahn-Banach theorem due to Rodé, Aequationes Math. 34 (1987), 89-95. Zbl0636.46005
- [10] P. Kranz, Additive functionals on abelian semigroups, Prace Mat. (Comment. Math.) 16 (1972), 239-246. Zbl0262.20087
- [11] J. Rätz, On approximately additive mappings, in: General Inequalities 2, Internat. Ser. Numer. Math. 47, Birkhäuser, Basel 1980, 233-251. Zbl0433.39014
- [12] G. Rodé, Eine abstrakte Version des Satzes von Hahn-Banach, Arch. Math. (Basel) 31 (1978), 474-481. Zbl0402.46003
- [13] L. Székelyhidi, Remark 17, Report of Meeting, the 22nd Internat. Symposium on Functional Equations, Aequationes Math. 29 (1985), 95-96.
- [14] J. Tabor, Remark 18, ibid., 96.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.