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

TL;DR

This paper characterizes when linear combinations of composition operators are compact on the Bloch space, linking compactness to the decay of certain operator norms involving iterates of the maps.

Contribution

It provides a necessary and sufficient condition for the compactness of linear combinations of composition operators on the Bloch space.

Findings

Compactness characterized by norm limits of iterated maps.

Condition involves the decay of the sum of iterates in the Bloch norm.

Extends analysis to the algebra of bounded analytic functions.

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.