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 \(L^p(\mathbb R)\), \(p>1\).

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 \(L^p(\mathbb R)\), \(p>1\).

Reviewer: Stefan Cobzaş (Cluj-Napoca)

##### 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 |