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

Let ${\lambda }_i(i=1,\mathrm{.}\mathrm{.}\mathrm{.},k)$ be any nonzero complex scalars and $\varphi_i (i=1,…,k)$beanyanalyticself-mapsoftheunitdisk$$\mathbb{D}$$. 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 $$\underset{n\to \mathrm{\infty }}{\lim }\mathrm{\parallel }{\lambda }_1{\phi }_1^n+{\lambda }_2{\phi }_2^n+\mathrm{.}\mathrm{.}\mathrm{.}+{\lambda }_k{\phi }_k^n{\mathrm{\parallel }}_{\mathcal{B}}=0.$$ We also study the linear combination of composition operators on the Banach algebra of bounded analytic functions.