TY - JOUR
T1 - Denseness of holomorphic functions attaining their numerical radii
AU - Acosta, María D.
AU - Kim, Sung Guen
PY - 2007/10
Y1 - 2007/10
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=58449098406&partnerID=8YFLogxK
U2 - 10.1007/s11856-007-0083-x
DO - 10.1007/s11856-007-0083-x
M3 - Article
AN - SCOPUS:58449098406
SN - 0021-2172
VL - 161
SP - 373
EP - 386
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
ER -