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Denseness of holomorphic functions attaining their numerical radii. (English) Zbl 1140.46007
For a complex Banach space X, let A (B X ;X) denote the space of bounded continuous functions h from B X ={xX:x1} to X which are holomorphic on {xX:x<1}. 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{|x * (h(x))|:xX, x * X * , x * =-x=x * (x)=1}. 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 (B X ;X) is dense in A (B X ;X). 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 (B c 0 ;c 0 ) 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(K) which are holomorphic on its interior. In the negative direction, the authors give an example of a certain Banach space (the pre-dual d * (ω,1) of the Lorentz space d(ω,1) for some ω in 2 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 {zd * (ω,1):z<R} (R>e) to d * (ω,1).

MSC:
46B99Normed linear spaces and Banach spaces
46B22Radon-Nikodým, Kreĭn-Milman and related properties
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