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 $\varphi_i (i=1,..,k)$ be any analytic self-maps of the unit disk $\mathbb{D}$. We show that the operator $\sum_{i=1}^k\lambda_iC_{\varphi_i}$ is compact on the Bloch space $\mathcal{B}$ if and only if $$\lim_{n\to\infty}|\lambda_1\varphi_1^n+\lambda_2\varphi_2^n+...+\lambda_k\varphi_k^n|_{\mathcal{B}}=0.$$ We also study the linear combination of composition operators on the Banach algebra of bounded analytic functions.