The authors study the topological space \({\mathcal C}(H^\infty)\) (endowed with the operator-norm topology) of composition operators \(C_\varphi\) on the space \(H^\infty\) of bounded analytic functions on the unit disk \(\mathbb{D}\). They answer a question of B. MacCluer, Sh. Ohno and R.-H. Zhao [Integral Equations Oper. Theory 40, No. 4, 481-494 (2001; Zbl 1062.47511)] by showing that \(C_\varphi\) is isolated in \({\mathcal C}(H^\infty)\) if and only if \(C_\varphi\) is essentially isolated. Recall that the essential semi-norm on \({\mathcal C}(H^\infty)\) is given by \[ \text{\(\|C_\varphi\|_e:=\inf\{\|C_\varphi-K\|: K\) is compact on \(H^\infty\}\)}. \] The proof uses their newly introduced notion of asymptotically interpolating sequence, a subject interesting in its own right [see P. Gorkin and R. Mortini, J. Lond. Math. Soc., II. Ser. 67, No. 2, 481-498 (2003)].
It is also shown that \(C_\varphi\) is isolated in \({\mathcal C}(H^\infty)\) if and only if \(\int_0^{2\pi} \log(1-|\varphi|) d\theta=-\infty\), that is iff \(\varphi\) is an extreme point of the unit ball in \(H^\infty\).