Shilov boundary for holomorphic functions on some classical Banach spaces

María D. Acosta; Mary Lilian Lourenço

Studia Mathematica, Tome 178 (2007), p. 27-39 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $_{\infty} (B_{X})$ 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 $_{u} (B_{X})$ be the subspace of $_{\infty} (B_{X})$ of those functions which are uniformly continuous on $B_{X}$ . A subset $B \subset B_{X}$ is a boundary for $_{\infty} (B_{X})$ if $∥ f ∥ = s u p_{x \in B} | f (x) |$ for every $f \in_{\infty} (B_{X})$ . 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 $_{\infty} (B_{X})$ . 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.

Publié le : 2007-01-01

Zbl 1124.46023

EUDML-ID : urn:eudml:doc:285309

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     author = {Mar\'\i a D. Acosta and Mary Lilian Louren\c co},
     title = {Shilov boundary for holomorphic functions on some classical Banach spaces},
     journal = {Studia Mathematica},
     volume = {178},
     year = {2007},
     pages = {27-39},
     zbl = {1124.46023},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm179-1-3}
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María D. Acosta; Mary Lilian Lourenço. Shilov boundary for holomorphic functions on some classical Banach spaces. Studia Mathematica, Tome 178 (2007) pp. 27-39. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm179-1-3/