TOPOLOGICAL PROPERTIES OF COMPOSITION OPERATORS ON SOME FRÉCHET ALGEBRA

Abstract

For let $F_{\beta }$ denote the Fréchet algebra of all holomorphic functions $f$ on the unit disk $\text{Δ}$ for which ${\lim }_{r\to 1}{(1-r)}^{\beta }{\log }^{+}\left[{\max }_{\left|z\right|\le r}\right|f\left(z\right)\left|\right]=0$. Given a holomorphic self-map $\varphi$ of $\text{Δ}$ define the composition operator $C_{\varphi }$ on $F_{\beta }$ by: $C_{\varphi }f=f\circ \varphi ,f\in F_{\beta }$. This note shows that $C_{\varphi }$ exists always as a continuous operator. Furthermore, this note points out that boundedness and compactness of $C_{\varphi }$ are not only the same, but also equivalent to ${\varphi }^n\text{exp}[r{n}^{\beta /(1+\beta )}]\to 0$ in $F_{\beta }$ for some .