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Density of a semigroup in a Banach space. (English. Russian original) Zbl 1316.46015
Izv. Math. 78, No. 6, 1079-1104 (2014); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 78, no. 6, 21-48 (2014).
For a nonempty subset $$M$$ of a Banach space $$X$$ denote by $$R(M)$$ the additive semigroup generated by $$M$$. The paper is concerned with the problem of the density of $$R(M)$$ in $$X$$, treated in two steps: I. one shows first that the closure $$\overline{R(M)}$$ of $$R(M)$$ is an additive subgroup of $$X$$, and II. one proves then that $$\overline{R(M)}=X.$$
A necessary condition for $$\overline{R(M)}$$ to be a subgroup is that the set $$M$$ is scalene, meaning that for every $$f\in X^*\setminus\{0\}$$ there exists $$x\in M$$ with $$f(x)<0$$ (resp. Re$$f(x)<0$$ in the complex case). In the finite-dimensional case this condition is also sufficient, and if $$M$$ is scalene and connected, then $$R(M)$$ is dense in $$X$$ (Theorem 1).
The set $$M$$ is called minimal if for every $$x\in M$$ and every neighborhood $$U(x)$$ of $$x$$ there exists $$f\in X^*$$ such that $$f(y)>0$$ (resp. Re $$f(y)>0$$) for all $$y\in M\setminus U(x)$$. The author gives an example of a scalene minimal connected compact set in each of the spaces $$L_2[0,1]$$ and $$L_1[0,1]$$ such that $$\overline{R(M)}$$ is not a subgroup. If the space $$X$$ is uniformly convex and $$\Gamma =\Gamma_1\cup\dots\cup\Gamma_m$$ is a scalene minimal subset of $$X$$ consisting of rectifiable Jordan curves $$\Gamma_j,$$ then $$\overline{R(M)}$$ is a subgroup of $$X$$ (Theorem 2). Under the same hypotheses, if $$X$$ is also uniformly smooth, then further $$\overline{R(M)}=X,$$ in the case $$m=1,$$ and it is a subgroup containing a linear subspace $$L$$ of codimension $$\leq m-1$$, in general (Theorem 6).
Applications are given to the problem of approximation in the space $$AC(K)$$ of functions $$f:K\to\mathbb X$$ continuous on $$K$$ and holomorphic in the interior of $$K$$, $$K$$ a compact subset of $$\mathbb C,$$ by sums of simple fractions of the form $$\sum_{k=1}^m(z-a_k)^{-1}$$ or $$\sum_{k=1}^n(z-a_k)^{-1}-m (z-b)^{-1}$$. As the author mentions, in fact it was this problem that motivated the general approach proposed in the paper. The paper contains also a brief survey on the approximation by sums of simple fractions in various Banach spaces of functions defined on various subsets of the complex plane; for instance, the fractions with poles in $$\mathbb C\setminus\mathbb R$$ are not dense in $$L^p(\mathbb R)$$, $$p>1$$.

##### MSC:
 46B20 Geometry and structure of normed linear spaces 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 46E15 Banach spaces of continuous, differentiable or analytic functions
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