Linear combination of composition operators on $H^\infty$ and the Bloch space

Abstract

Let $\lambda _i (i=1,\ldots ,k)$ be nonzero complex scalars and $\varphi _i (i=1,..,k)$ be analytic self-maps of the unit disk $\mathbb {D}$. We show that the operator $\sum _{i=1}^k\lambda _iC_{\varphi _i}$ is compact on the Bloch space $\mathcal {B}$ if and only if

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert \lambda _1\varphi _1^n+\lambda _2\varphi _2^n+\cdots +\lambda _k\varphi _k^n\Vert _{\mathcal {B}}=0. \end{aligned}$$

We also study the linear combination of composition operators on the Banach algebra of bounded analytic functions.