Shilov boundary for holomorphic functions on some classical Banach spaces

• Autorzy: María D. Acosta , Mary Lilian Lourenço
• Języki publikacji: EN

Abstrakty

EN

Let $𝓐_{∞}(B_{X})$ 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 $𝓐_{u}(B_{X})$ be the subspace of $𝓐_{∞}(B_{X})$ of those functions which are uniformly continuous on $B_{X}$. A subset $B ⊂ B_{X}$ is a boundary for $𝓐_{∞}(B_{X})$ if $∥f∥ = sup_{x∈ B} |f(x)|$ for every $f ∈ 𝓐_{∞}(B_{X})$. We prove that for X = d(w,1) (the Lorentz sequence space) and X = C₁(H), the trace class operators, there is a minimal closed boundary for $𝓐_{∞}(B_{X})$. On the other hand, for X = 𝓢, the Schreier space, and $X = K(ℓ_{p},ℓ_{q})$ (1 ≤ p ≤ q < ∞), there is no minimal closed boundary for the corresponding spaces of holomorphic functions.

Kategorie tematyczne

• 46B99: None of the above, but in this section
• 46J20: Ideals, maximal ideals, boundaries
• 46G20: Infinite-dimensional holomorphy

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2007
• Tom: 179
• Numer: 1
• Strony: 27-39

Daty

wydano

2007

Twórcy

autor

María D. Acosta

• Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

autor

Mary Lilian Lourenço

• Departamento de Matemática e Estatística, Universidade de São Paulo, CP 66281 CEP, 05311-970 São Paulo, Brazil

Identyfikatory

DOI

10.4064/sm179-1-3