Convexity and polynomial equations in Banach spaces with Radon-Nikodym property. (Q1812210)

Let \((\Omega,\Sigma,\mu)\) be a finite measure space, let \(S(\mu)\) denote the subspace of \(L_1(\Omega,\Sigma,\mu)\) consisting of simple functions, and let \(\mathbb{R}[x]\) denote the space of polynomials. The polynomials in question are elements \(P \in R[x] \otimes S(\mu),\) that is all functions of the form \(P(x) = \sum_{j=1}^m P_j(x) \otimes f_j\) where each \(P_j \in \mathbb{R}[x]\) and \(f_j \in S(\mu).\) The annihilator of such a \(P\) is by definition \(An(P) = \{f \in L_1(\mu) \;| \;P(f) = 0 \}.\) The author's main result is the following theorem, which may be viewed as a generalization of Theorem IX.1.10 of J. J. Uhl [\textit{J. Diestel} and \textit{J. J. Uhl, jun.}, ``Vector measures'', AMS Math. Surveys 15 (1977; Zbl 0369.46039)]: Let \(E\) be a Banach space having the Radon-Nikodym property, let \(P\) be a polynomial that is not a divisor of zero, and let \(F:\Sigma \to E\) be a countably additive vector measure of bounded variation which is absolutely continuous with respect to \(\mu.\) If \(T_F:S(\mu) \to E\) is the integration map, \(f \in S(\mu) \mapsto \int f\, dF,\) then the norm closure of \(T_F(An(P))\) is convex and compact. Several consequences are obtained, related to approximable subsets of \(L_1(\mu).\) We say that \(K \subset L_1(\mu)\) is approximable by polynomials if for each \(\epsilon > 0,\) there is a polynomial \(P\) that is not a zero divisor such that for every \(f \in K,\) there is \(f_0 \in An(P)\) satisfying \(| | f - f_0 | | < \varepsilon.\) In addition, if for every \(\varepsilon > 0\) there is a polynomial \(P\) that is not a zero divisor such that for every \(f_0 \in An(P),\) there is \(f \in K\) satisfying \(| | f - f_0 | | < \varepsilon,\) then \(K\) is called strictly approximable by polynomials. The author proves that compact subsets of \(L_1(\mu)\) are approximable by polynomials, while every such set is relatively weakly compact. Furthermore, given a subset \(K\) of \(L_1(\mu)\) that is approximable by polynomials and an arbitrary operator \(T:L_1(\mu) \to E,\) \(T(K)\) is relatively compact. Moreover, if \(K\) is strictly approximable by polynomials, then \(\overline{T(K)}\) is convex. The paper concludes with a number of open problems.