Composition operators on vector-valued analytic function spaces: a survey (Q2875927)

This nice, informative survey reviews recent results about composition operators on various Banach spaces of vector valued analytic functions, both of weak and strong type. The authors highlight new phenomena that do not have any counterparts for scalar valued composition operators. Boundedness and (weak) compactness of vector valued composition operators on the Hardy space \(H^p(X)\), \(1 \leq p \leq \infty\), \(X\) a Banach space, were first systematically investigated by \textit{P.-D. Liu} et al. [Pac. J. Math. 184, No. 2, 295--309 (1998; Zbl 0932.47023)]. Section 2 introduces a general, flexible framework to study qualitative properties of vector valued composition operators. Weak compactness on the strong vector valued Hardy space \(H^1(X)\) is studied in Section 3. A more general case is also presented and it is applied to weighted Bergman spaces \(A^{\alpha}_p(X)\), \(\alpha>-1\), \(1 \leq p < \infty\), weighted Banach spaces \(H^{\infty}_v(X)\), the vector valued Bloch space \(\mathcal{B}(X)\), vector valued Cauchy transforms, etc. Composition operators on vector valued BMOA-spaces are investigated n Section 4. Here, several interesting results by the first author et al. [Complex Anal. Oper. Theory 7, No. 1, 163--181 (2013; Zbl 1295.47019)] are presented. Weak spaces of vector valued analytic functions and composition operators between them were introduced and studied by \textit{P. Domański}, \textit{M. Lindström} and the reviewer in [Ann. Acad. Sci. Fenn., Math. 26, No. 1, 233--248 (2001; Zbl 1075.47506)]. Several results about weakly compact composition operators, a linearization theorem and a comparison between weak and strong spaces, including results due to the present authors in [Indiana Univ. Math. J. 55, No. 2, 719--746 (2006; Zbl 1119.47022)], are presented in Section 5. Wang and the authors initiated in [J. Oper. Theory 62, No. 2, 281--295 (2009; Zbl 1199.47110)], the investigation of properties of composition operators from weak to strong spaces. Some theorems in this direction are reviewed in Section 6. In the final Section 7, results about operator weighted composition operators between two spaces of type \(H^{\infty}_v(X)\), due to the authors [Ill. J. Math. 53, No. 4, 1019--1032 (2009; Zbl 1207.47021)], are reviewed. Extensions to the locally convex setting are due to Manhas and to Gómez-Collado, Jornet, Elke Wolf and the reviewer. Several interesting open problems are formulated in the paper.