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Denseness of holomorphic functions attaining their numerical radii. (English) Zbl 1140.46007
For a complex Banach space $X$, let ${A}_{\infty }\left({B}_{X};X\right)$ denote the space of bounded continuous functions $h$ from ${B}_{X}=\left\{x\in X:\parallel x\parallel \le 1\right\}$ to $X$ which are holomorphic on $\left\{x\in X:\parallel x\parallel <1\right\}$. The numerical radius of $h$ was first defined by L. A. Harris in [Am. J. Math. 93, 1005–1019 (1971; Zbl 0237.58010)] as the quantity $sup\left\{|{x}^{*}\left(h\left(x\right)\right)|:x\in X$, ${x}^{*}\in {X}^{*}$, $\parallel {x}^{*}\parallel =-\parallel x\parallel ={x}^{*}\left(x\right)=1\right\}$. $h$ is said to attain its numerical radius if this supremum is attained for some $x$ and ${x}^{*}$. The present paper discusses when the subset of the numerical radius attaining elements in ${A}_{\infty }\left({B}_{X};X\right)$ is dense in ${A}_{\infty }\left({B}_{X};X\right)$. On the positive side, it is shown that this is the case when $X$ has the Radon–Nikodým property. For some classical spaces without this property, there are also obtained some positive results. For example, the denseness of the numerical radius attaining elements in ${A}_{\infty }\left({B}_{{c}_{0}};{c}_{0}\right)$ follows from the previously known denseness of norm attaining elements. For $K$ a compact Hausdorff topological space, the numerical radius attaining elements are shown to be dense in the space of bounded weakly uniformly continuous functions on ${B}_{C\left(K\right)}$ which are holomorphic on its interior. In the negative direction, the authors give an example of a certain Banach space (the pre-dual ${d}_{*}\left(\omega ,1\right)$ of the Lorentz space $d\left(\omega ,1\right)$ for some $\omega$ in ${\ell }^{2}\setminus {\ell }^{1}$)) for which the subset of the numerical radius attaining elements is not dense in the Banach space of all bounded holomorphic functions from the open subset $\left\{z\in {d}_{*}\left(\omega ,1\right):\parallel z\parallel $\left(R>e\right)$ to ${d}_{*}\left(\omega ,1\right)$.

##### MSC:
 46B99 Normed linear spaces and Banach spaces 46B22 Radon-Nikodým, Kreĭn-Milman and related properties
##### References:
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