Isolation amongst the composition operators
Let \(H^ 2\) denote the Hardy space of holomorphic functions on the unit disc U in \({\mathbb{C}}\). If \(\phi\) is a holomorphic self-map of U, \(C_{\phi}\) denotes the composition operator on \(H^ 2\) defined by \(C_{\phi}(f)=f\circ \phi\). \textit{E. Berkson} [Proc. Am. Math. Soc. 81, 230-232 (1981; Zbl 0464.30027)] has shown that certain highly non-compact composition operators are isolated amongst the composition operators on \(H^ 2\) in the operator norm topology. On the other hand, as is shown in the paper, no compact composition operator is isolated. In the paper, the authors explore the intermediate territory. It is shown that if \(\phi\) satisfies \[ (*)\quad \int^{2\pi}_{0}\log (1-| \phi (e^{i\theta})|)d\theta >-\infty, \] then \(C_{\phi}\) is not isolated among the composition operators on \(H^ 2\). As a consequence, only the extreme points of the unit ball of \(H^{\infty}\) can induce isolated composition operators. It is also shown that condition (*), though sufficient for non-isolation, not necessary: there exist extreme points which induce compact composition operators, which are therefore not isolated. In the other direction, it is shown that whenever \(\phi\) is a univalent extreme point mapping U onto a sufficiently regular sub- region, then \(C_{\phi}\) is isolated.
scientific article; zbMATH DE number 4210501