Denseness of holomorphic functions attaining their numerical radii

María D. Acosta, Sung Guen Kim

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

For two complex Banach spaces X and Y, A-∞ (B X; Y) will denote the space of bounded and continuous functions from B X to Y that are holomorphic on the open unit ball. The numerical radius of an element h in A (B X; X) is the supremum of the set |x*(h(x))| : x ε X, x x* ε X x*, || x x* parallel = x = x* (x) = 1. We prove that every complex Banach space X with the Radon-Nikodým property satisfies that the subset of numerical radius attaining functions in A (B X; X) is dense in {A} (B X; X). We also show the denseness of the numerical radius attaining elements of Au (Bc0 } ; c 0 ) in the whole space, where Bc0 } ; c0 is the subset of functions in Bc0 } ; c0 ) which are uniformly continuous on the unit ball. For C(K) we prove a denseness result for the subset of the functions in {A (B C(K); C(K)) which are weakly uniformly continuous on the closed unit ball. For a certain sequence space X, there is a 2-homogenous polynomial P from X to X such that for every R > e, P cannot be approximated by bounded and numerical radius attaining holomorphic functions defined on RB X . If Y satisfies some isometric conditions and X is such that the subset of norm attaining functions of A (B X; ℂ) is dense in A} Bc0 } ; c0 (B X; ℂ), then the subset of norm attaining functions in {A} (B X; Y) is dense in the whole space.

Original languageEnglish
Pages (from-to)373-386
Number of pages14
JournalIsrael Journal of Mathematics
Volume161
DOIs
StatePublished - Oct 2007

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