Shilov boundary for holomorphic functions on some classical Banach spaces

María D. Acosta, and Mary Lilian Lourenço. "Shilov boundary for holomorphic functions on some classical Banach spaces." Studia Mathematica 179.1 (2007): 27-39. <http://eudml.org/doc/285309>.

@article{MaríaD2007,
abstract = {Let $_\{∞\}(B_\{X\})$ 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 $_\{u\}(B_\{X\})$ be the subspace of $_\{∞\}(B_\{X\})$ of those functions which are uniformly continuous on $B_\{X\}$. A subset $B ⊂ B_\{X\}$ is a boundary for $_\{∞\}(B_\{X\})$ if $∥f∥ = sup_\{x∈ B\} |f(x)|$ for every $f ∈ _\{∞\}(B_\{X\})$. 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 $_\{∞\}(B_\{X\})$. On the other hand, for X = , 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.},
author = {María D. Acosta, Mary Lilian Lourenço},
journal = {Studia Mathematica},
keywords = {spaces of holomorphic functions; boundary; Shilov boundary; peak point; strong peak point; Schreier space; space of compact operators; space of trace class operators; Lorentz sequence space},
language = {eng},
number = {1},
pages = {27-39},
title = {Shilov boundary for holomorphic functions on some classical Banach spaces},
url = {http://eudml.org/doc/285309},
volume = {179},
year = {2007},
}

TY - JOUR
AU - María D. Acosta
AU - Mary Lilian Lourenço
TI - Shilov boundary for holomorphic functions on some classical Banach spaces
JO - Studia Mathematica
PY - 2007
VL - 179
IS - 1
SP - 27
EP - 39
AB - Let $_{∞}(B_{X})$ 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 $_{u}(B_{X})$ be the subspace of $_{∞}(B_{X})$ of those functions which are uniformly continuous on $B_{X}$. A subset $B ⊂ B_{X}$ is a boundary for $_{∞}(B_{X})$ if $∥f∥ = sup_{x∈ B} |f(x)|$ for every $f ∈ _{∞}(B_{X})$. 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 $_{∞}(B_{X})$. On the other hand, for X = , 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.
LA - eng
KW - spaces of holomorphic functions; boundary; Shilov boundary; peak point; strong peak point; Schreier space; space of compact operators; space of trace class operators; Lorentz sequence space
UR - http://eudml.org/doc/285309
ER -