On compactness of the difference of composition operators
Let \(\phi\) and \(\psi\) be analytic self-maps of the unit disc, and denote by \(C_\phi\) and \(C_\psi\) the induced composition operators. The authors study the compactness of the difference operator \(T=C_\phi-C_\psi\) on the Hardy spaces \(H^p\), the Lebesgue spaces \(L^p\), and the measure space \(M\) of the unit circle. The authors prove several results described below. 1. (Compactness on \(H^p\)) It is shown that the compactness of \(T\) on \(H^p\) is independent of \(p\in[1,\infty)\) and equivalent to the weak compactness on \(H^1\). However, the authors do not provide any characterization in terms of symbols. 2. (Compactness on \(L^1\) and \(M\)) It is shown that \(T\) is either compact or weakly compact on \(L^1\) if and only if \(T\) is either compact or weakly compact on \(M\). Also, a characterization in terms of the Aleksandrov measures of symbols is provided. 3. (Different nature of \(H^1\) and \(L^1\)) It is well-known that the compactness of a single composition operator on \(H^1\) is equivalent to the compactness on \(L^1\) or \(M\). It is shown that this phenomenon does not extend to difference operators. The authors construct an explicit counterexample \(T\) which is compact on \(H^1\), but not on \(L^1\). The construction is substantially complicated and relies on somewhat delicate estimates involving harmonic measures. 4. The authors construct another example of a non-compact operator \(T\) on \(H^1\) for which the singular parts of the Aleksandrov measures of symbols coincide everywhere on the unit circle. This shows that there is no redundant condition in the Aleksandrov measure characterization of compactness on \(L^1\) mentioned above. In addition, this example disproves a conjecture posed by \textit{J. E. Shapiro} [J. Oper. Theory 40, 133--146 (1998; Zbl 0997.47023)]. 5. (Compactness on \(H^\infty\) and \(L^\infty\)) \(T\) is either compact or weakly compact on \(H^\infty\) if and only if \(T\) is either compact or weakly compact on \(L^\infty\). This extends an earlier characterization of the compactness of \(T\) on \(H^\infty\) given by \textit{B. McCluer, S. Ohno} and \textit{R. Zhao} [Integral Equations Oper. Theory 40, 481--494 (2001; Zbl 1062.47511)].
scientific article; zbMATH DE number 2107281
scientific article