Holomorphic functions and polynomial ideals on Banach spaces

DANIEL CARANDO; VERÓNICA DIMANT; SANTIAGO MURO

COLLECTANEA MATHEMATICA

UNIV BARCELONA

Año: 2012 vol. 63 p. 71 - 91

0010-0757

Given $\u$ a multiplicative sequence of polynomial ideals, weconsider the associated algebra of holomorphic functions of boundedtype, $H_{b\u}(E)$.\ We prove that, under very natural conditionssatisfied by many usual classes of polynomials, the spectrum$M_{b\u}(E)$ of this algebra ``behaves'' like the classical case of$M_{b}(E)$ (the spectrum of $H_b(E)$, the algebra of bounded typeholomorphic functions). More precisely, we prove that $M_{b\u}(E)$can be endowed with a structure of  Riemann domain over~$E''$ andthat the extension of each $f\in H_{b\u}(E)$ to the spectrum is an$\u$-holomorphic function of bounded type in each connectedcomponent.\ We also prove a Banach-Stone type theorem for thesealgebras.