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Shilov boundary for holomorphic functions on some classical Banach spaces

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Acosta M., Lourenço M.

Studia Mathematica

2007

tom 179

nr 1

27-39

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.

Ograniczanie wyników

1 Studia Mathematica

1 Acosta M.

1 Lourenço M.

1 2007

Wyniki wyszukiwania

Shilov boundary for holomorphic functions on some classical Banach spaces