Numerical ranges of uniformly continuous functions on the unit sphere of a Banach space
Given a Banach space \(X\), its unit sphere is denoted by \(S_X\), its closed unit ball is \(B_X\), and \(\ell_\infty(S_X,X)\) denotes the Banach space of all bounded functions from \(S_X\) to \(X\). The topological dual of \(X\) is \(X^*\) and \(\Pi(X)\) stands for the set of those couples \((x,\phi)\in S_X \times S_{X*}\) satisfying \(\phi(x)=1\). Given \(f\in \ell_\infty(S_X,X)\), its {spatial numerical range} is the set \[ W(f):=\{\phi(f(x)) : \;(x,\phi)\in \Pi(X)\}, \] and its {intrinsic numerical range} is \[ V(f):=\{\Phi(f) : \;\Phi\in \ell_\infty(S_X,X)^*,\;\| \Phi\| =1,\;\Phi(\text{Id})=1\}. \] Note that the inclusion \[ \overline{\text{co}}\;W(f) \subseteq V(f) \tag{1} \] (\(\overline{\text{co}}\) means closed convex hull) holds for every \(f\in \ell_\infty(S_X,X)\), but the equality holds for every \(f\in \ell_\infty(S_X,X)\) (if and) only if \(X\) is uniformly smooth [\textit{A. Rodríguez-Palacios}, Proc. Am. Math. Soc. 129, No. 3, 815--821 (2001; Zbl 0969.46011)]. The aim of the paper under review is to show that equality in (1) holds in an improved way whenever \(f:S_X\to X\) is bounded and uniformly continuous. Actually, Theorem~2.5 states the following: for every subset \(\Gamma\) of \(\Pi(X)\) such that the natural projection \(\pi_X(\Gamma)\) is dense in \(S_X\), the equality \[ V(f)=\overline{\text{co}}\;\{\phi(f(x))\;: \;(x,\phi)\in \Gamma\} \tag{2} \] holds for every bounded and uniformly continuous function \(f:S_X\to X\). Let us mention that some particular cases of the above result have already been known. It was proved in [\textit{F. F. Bonsall} and \textit{J. Duncan}, Numerical ranges of operators on normed spaces and of elements of normed algebras (Cambridge University Press, London) (1971; Zbl 0207.44802), Theorem~9.3] that equality (2) holds for (the restriction to \(S_X\) of) any bounded linear operator on \(X\). In the complex case, it was proved in [\textit{L. A. Harris}, Am. J. Math. 93, 1005--1019 (1971; Zbl 0237.58010)] that equality \((2)\) holds for every uniformly continuous function from \(B_X\) to \(X\) which is holomorphic in the open unit ball, provided that \(\pi_X(\Gamma)=S_X\). Finally, in the case \(\Gamma=\Pi(X)\), equality \((2)\) was proved in [\textit{L. A. Harris}, Proc. Camb. Philos. Soc. 76, 133--141 (1974; Zbl 0303.41029)] for arbitrary bounded and uniformly continuous functions from \(S_X\) to \(X\). Anyhow, the techniques used in the present paper are different from the ones used in such particular cases.
scientific article; zbMATH DE number 2107199
scientific article