Compact linear combinations of composition operators on Hardy spaces

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