The algebra of bounded-type holomorphic functions on the ball

Carando, Daniel; Muro, Santiago; Vieira, Daniela

doi:10.1090/proc/14471

The algebra of bounded-type holomorphic functions on the ball
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by Daniel Carando, Santiago Muro and Daniela M. Vieira PDF

Proc. Amer. Math. Soc. 148 (2020), 2447-2457 Request permission

Abstract:

We study the spectrum $M_b(U)$ of the algebra of bounded-type holomorphic functions on a complete Reinhardt domain in a symmetrically regular Banach space $E$ as an analytic manifold over the bidual of the space. In the case that $U$ is the unit ball of $\ell _p$, $1<p<\infty$, we prove that each connected component of $M_b(B_{\ell _p})$ naturally identifies with a ball of a certain radius. We also provide estimates for this radius and in many natural cases we have the precise value. As a consequence, we obtain that for connected components different from that of evaluations, these radii are strictly smaller than one, and can be arbitrarily small. We also show that for other Banach sequence spaces, connected components do not necessarily identify with balls.

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