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

Abstract

Let ${\lambda }_i(i=1,...,k)$ be any nonzero complex scalars and ${\phi }_i(i=1,..,k)$ be any analytic self-maps of the unit disk $\mathbb{D}$. We show that the operator ${\sum }_{i=1}^k{\lambda }_i{C}_{{\phi }_i}$ is compact on the Bloch space $\mathcal{B}$ if and only if $$\underset{n\to \infty }{\lim }\Vert {\lambda }_1{\phi }_1^n+{\lambda }_2{\phi }_2^n+...+{\lambda }_k{\phi }_k^n{\Vert }_{\mathcal{B}}=0.$$ We also study the linear combination of composition operators on the Banach algebra of bounded analytic functions.