Cluster values of holomorphic functions of bounded type

Aron, Richard; Carando, Daniel; Lassalle, Silvia; Maestre, Manuel

doi:10.1090/tran/6407

Cluster values of holomorphic functions of bounded type
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by Richard M. Aron, Daniel Carando, Silvia Lassalle and Manuel Maestre PDF

Trans. Amer. Math. Soc. 368 (2016), 2355-2369 Request permission

Abstract:

We study the cluster value theorem for $H_b(X)$, the Fréchet algebra of holomorphic functions bounded on bounded sets of $X$. We also describe the (size of) fibers of the spectrum of $H_b(X)$. Our results are rather complete whenever $X$ has an unconditional shrinking basis and for $X=\ell _1$. As a byproduct, we obtain results on the spectrum of the algebra of all uniformly continuous holomorphic functions on the ball of $\ell _1$.

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