Compact linear combinations of composition operators on Hardy spaces

Let \(\varphi _j\) ${\phi }_j$ , \(j=1,2, \ldots , N\) $j=1,2,\dots ,N$ , be holomorphic self-maps of the unit disk \({\mathbb {D}}\) $\mathbb{D}$ of \({\mathbb {C}}\) $\mathbb{C}$ . We prove that the compactness of a linear combination of the composition operators \(C_{\varphi _j}: f\mapsto f\circ \varphi _j\) $C_{{\phi }_j}:f\mapsto f\circ {\phi }_j$ on the Hardy space \(H^p({\mathbb {D}})\) $H^p(\mathbb{D})$ does not depend on p for \(0<p<\infty \) $0<p<\infty$ . This answers a conjecture of Choe et al. about the compact differences \(C_{\varphi _1} - C_{\varphi _2}\) $C_{{\phi }_1}-C_{{\phi }_2}$ on \(H^p({\mathbb {D}})\) $H^p(\mathbb{D})$ , \(0<p<\infty \) $0<p<\infty$ .