A characterization of maximal ideals in the Fr\'{e}chet algebras of holomorphic functions $F^p$ $(1<p<\infty$

Abstract

The space $F^p$ ($1<p<\infty$) consists of all holomorphic functions $f$ on the open unit disk $\Bbb D$ for which $\lim_{r\to 1}(1-r)^{1/q}\log^+M_{\infty}(r,f)=0,$ where $M_{\infty}(r,f)=\max_{\vert z\vert\le r}\vert f(z)\vert$ with $0<r<1$. Stoll [5, Theorem 3.2] proved that the space $F^p$ with the topology given by the family of seminorms $\left\{\Vert \cdot\Vert_{q,c}\right\}_{c>0}$ defined for $f\in F^q$ as $\Vert f\Vert_{q,c}:=\sum_{n=0}^{\infty}\vert a_n\vert\exp\left(-cn^{1/(q+1)} \right)<\infty,$ is a countably normed Fr\'{e}chet algebra. Notice that for each $p>1$, $F^p$ is the Fr\'{e}chet envelope of the Privalov space $N^p$. In this paper we study the structure of maximal ideals in the algebras $F^p$ ($1<p<\infty$). In particular, we give a complete characterization of closed maximal ideals in $F^p$. Moreover, we characterize multiplicative linear functionals on $F^p$.