Compact and weakly compact composition operators on BMOA (Q371772)

The authors prove that the weak compactness and compactness for composition operators \(C_\phi\) acting on BMOA are equivalent. There are already different characterizations of compactness of \(C_\phi\) on BMOA. The authors improve the one using the counting Nevanlinna function due to \textit{W. Smith} [Proc. Am. Math. Soc. 127, No. 9, 2715--2725 (1999; Zbl 0921.47025)] who showed that \(C_\phi\) is compact on BMOA if and only if \(\phi\) satisfies two conditions, which are shown not to be independent in this paper. In fact, one follows from the other one and hence the compactness of \(C_\phi\) on BMOA can now be described simply by the condition \(\lim_{|\phi(a)|\to 1}\|\sigma_{\phi(a)}\circ\phi\circ \sigma_a\|_{H^2}=0\), where \(\sigma_b\) denotes the standard Möbius transformation with \(\sigma_b(0)=b\). They complete their main result by establishing the equivalence with the weak compactness on BMOA. For the operators \(C_\phi\) acting on VMOA, the situation is also completely solved showing that in this case it also happens to be true that \(C_\phi\) is compact and weakly compact on VMOA are equivalent (solving a question raised by \textit{P. S. Bourdon} et al. in [Trans. Am. Math. Soc. 351, No. 6, 2183--2196 (1999; Zbl 0920.47029)]) and are characterized by the condition \(\lim_{|a|\to 1}\|\sigma_{\phi(a)}\circ\phi\circ \sigma_a\|_{H^2}=0\). A rather interesting corollary of several formulations of this description leads to that the compactness of \(C_\phi\) on VMO is equivalent also to the compactness of \(C_\phi\) as operator from \(\mathcal B\) to VMOA.