Shilov boundary for holomorphic functions on some classical Banach spaces

María D. Acosta ¹   ; Mary Lilian Lourenço ²

¹ Departamento de Análisis Matemático Facultad de Ciencias Universidad de Granada 18071 Granada, Spain
² Departamento de Matemática e Estatística Universidade de São Paulo CP 66281 CEP 05311-970 São Paulo, Brazil

Studia Mathematica, Tome 179 (2007) no. 1, pp. 27-39

Résumé

Let $ {\mathcal{A}}_\infty (B_X)$ be the Banach space of all bounded and continuous functions on the closed unit ball $B_X$ of a complex Banach space $X$ and holomorphic on the open unit ball, with sup norm, and let $ {\mathcal{A}}_{\rm u} (B_X)$ be the subspace of $ {\mathcal{A}}_\infty (B_X)$ of those functions which are uniformly continuous on $B_X.$ A subset $B \subset B_X$ is a boundary for $ {\mathcal{A}}_\infty (B_X)$ if $\Vert f \Vert = \sup _{ x \in B} \vert f(x) \vert $ for every $f \in {\mathcal{A}}_\infty (B_X)$. We prove that for $X= d(w,1) $ (the Lorentz sequence space) and $X= C_1(H)$, the trace class operators, there is a minimal closed boundary for $ {\mathcal{A}}_\infty (B_X)$. On the other hand, for $X=\mathcal{S}$, the Schreier space, and $X= K(\ell_p, \ell_q ) $ ($1 \le p \le q \infty$), there is no minimal closed boundary for the corresponding spaces of holomorphic functions.

Détail

DOI : 10.4064/sm179-1-3

Keywords: mathcal infty banach space bounded continuous functions closed unit ball complex banach space holomorphic unit ball sup norm mathcal subspace mathcal infty those functions which uniformly continuous subset subset boundary mathcal infty vert vert sup vert vert every mathcal infty prove lorentz sequence space trace class operators there minimal closed boundary mathcal infty other mathcal schreier space ell ell infty there minimal closed boundary corresponding spaces holomorphic functions