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Composition operators on vector-valued harmonic functions and Cauchy transforms. (English) Zbl 1119.47022
Let $\varphi$ be an analytic map of the unit disk $\Bbb{D}$ into itself, and denote by $h^1$, $H^1$, $PL^1$, and $CT$ the harmonic Hardy space of the unit circle $T$, the analytic Hardy space, the space of Poisson integrals of $L^1(\Bbb{T},dm)$ functions, and the space of Cauchy transforms of measures on the unit circle, respectively. The composition operator $C\sb\varphi$, defined by $C_{\varphi} f = f\circ\varphi$, is bounded on each of these spaces. The following well-known theorem was obtained by combining the efforts of D. Sarason with those of J. H. Shapiro and C. Sundberg. The operator $C\sb\varphi$ is compact on $h\sp1$ if and only if it is compact on $PL\sp1$, if and only if it is such on $H\sp1$; any of these conditions is further equivalent to $$ (\ast) \qquad \int_{\Bbb{T}} {1 -\vert \varphi(\xi))\vert ^2 \over \vert \zeta - \varphi (\xi)\vert ^2} d m (\xi) = \text{Re}\,\left( {\zeta +\varphi(0) \over \zeta - \varphi(0)} \right) \qquad \text{ for all} \quad \zeta\in \Bbb{T}, $$ as well as to the weak compactness on any of the spaces mentioned. Here, $\varphi$ actually denotes the radial limit function of $\varphi$, defined almost everywhere on $\Bbb{T}$. (Related partial results also exist for the space $CT$ and are due to P. Bourdon, J. Cima, and A. Matheson.) Inspired by the growing interest in composition operators in a vector-valued setting, as a main result, the authors of the paper under review extend the above characterization to a more general context. Namely, it is possible to define $C\sb\varphi$ on a complex Banach space $X$ and the functions $f\colon \Bbb{D}\to X$, as well as the vector-valued harmonic Hardy space $h\sp 1(X)$ and the space $CT(X)$ of vector-valued Cauchy transforms. The authors prove in this nice paper that $C\sb\varphi$ is weakly compact on either space $h\sp 1(X)$, $CT(X)$ if and only if the above condition ($\ast$) holds and $X$ is reflexive. (It should be remarked that the operator cannot be compact as long as $X$ is infinite-dimensional.) The difficult part is to prove the statement under the assumption that $X$ is reflexive. The authors employ a clever argument which allows them to avoid discussing the existence of radial limits of the functions in $h\sp p(X)$, among other numerous technical tools, many of them of abstract functional-analytic nature. The paper also extends and complement recent results by Bonet, Domański, and Lindström and by Blasco for composition operators on weak spaces $w H\sp p(X)$ and $w h\sp p(X)$ of vector-valued harmonic functions. Examples are also provided that show that for any complex infinite-dimensional Banach space $X$ and $1\le p<\infty$, there exist functions which are in $w H\sp p(X)$ but not in $H\sp p(X)$ (resp., in $w h\sp p(X)$ but not in $w h\sp p(X)$). The paper also contains several other interesting results, examples and comments.

47B33Composition operators
46E40Spaces of vector- and operator-valued functions
46G10Vector-valued measures and integration
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