Compact linear combinations of composition operators on Hardy spaces

Abstract

Let $\ddot{I}{†}_j$, $j=1,2,\dots ,N$, 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 $C_{\ddot{I}{†}_j}:f\mapsto f\circ \ddot{I}{†}_j$ 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 $C_{\ddot{I}{†}_1}-C_{\ddot{I}{†}_2}$ on $H^p(\mathbb{D})$, $0<p<\infty$.

6 pages. The order of presentation in Sections 2 and 3 is changed