ISSN:
1420-8903
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract. We consider the following problems: Let $(S, +)$ be a (not necessarily commutative) semigroup and let $\rho : S \to {\Bbb R}$ be a given “control” functional (which is not assumed to be sub-additive). Assume that $F : S \to {\Bbb R}$ is a “nearly additive” mapping in either of the following ways: (1) $|F(x + y) - F(x) - F(y)| \le \rho(x) + \rho(y) - \rho(x + y)$ holds for all $x, y \in S$ . (2) $\Bigl|\sum^n_{i = 1} F(x_i) - \sum^m_{j = 1} F(y_j)\Bigr| \le \sum^n_{i = 1} \rho(x_i) + \sum^m_{j = 1} \rho(y_j) $ holds for all n and m whenever $x_i$ and $y_j$ are elements of S such that $\sum^n_{i = 1} x_i = \sum^m_{j = 1} y_j$ . Must F be “near” to an additive mapping $A : S \to {\Bbb R}$ in the sense that (3) $|F(x) - A(x)| \le K\rho (x)$ for some K and all $x \in S$ ? We prove Theorem. Let $(S, +)$ be a semigroup and let K be a fixed number. The following statements are equivalent: (a) For every ρ and every F satisfying (1) there is an additive A fulfilling (3). (b) For every ρ and every F satisfying (2) there is an additive A fulfilling (3). Moreover, for K = 1 both (a) and (b) are equivalent to (c) For every $\alpha, \beta : S \to {\Bbb R}$ such that α is superadditive (i.e., $\alpha(x + y) \ge \alpha(x) + \alpha(y))$ , β is subadditive and $\alpha \le \beta$ , there exists an additive A separating α from β (that is, satisfying $a(x) \le A(x) \le \beta (x))$ . Combining this theorem with previous results of Ger and Gajda and Kominek we obtain that every weakly commutative group satisfies (b) for K = 1 and every amenable group satisfies (b) for K = 2, which gives a partial answer to a problem suggested by Forti, improving results by Šemrl and Castillo and the present author. We close the paper with some remarks about vector-valued mappings and the connections between “nearly additive” mappings and the theory of extensions of Banach spaces.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s000100050059
