# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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.

##### MSC:
 47B33 Composition operators 46E40 Spaces of vector- and operator-valued functions 46G10 Vector-valued measures and integration
Full Text: