zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Weighted composition operators between Bloch-type spaces. (English) Zbl 1042.47018
For analytic functions $u$ on the unit disk $D$ and analytic mappings $\phi: D \to D$, the weighted composition operator $uC_\phi$ is defined by $uC_\phi(f) = u(f \circ \phi)$ for $f$ analytic on $D$. In the paper under review, the authors consider these operators acting on the weighted Bloch-type spaces $\Bbb B^\alpha$ and $\Bbb B^\alpha_0$, $ 0 < \alpha < \infty$, defined by $$\Bbb B^\alpha = \{f \in H(D): \sup_{z\in D} (1 -\vert z\vert ^2)^\alpha \vert f'(z)\vert < \infty\}$$ and $$\Bbb B^\alpha_0 =\{f \in \Bbb B^\alpha : \lim_{\vert z\vert \to 1} (1 - \vert z\vert ^2)^\alpha \vert f'(z)\vert = 0\}.$$ The main results completely characterize boundedness and compactness of $uC_\phi$ from $\Bbb B^\alpha$ to $\Bbb B^\beta$ as well as from $\Bbb B^\alpha_0$ to $\Bbb B^\beta_0$. Finally, the authors give some examples of functions $u$ and $\phi$ for which $uC_\phi$ between the various spaces is bounded, compact or noncompact. Similar results were obtained by {\it M. D. Contreras} and {\it A. G. Hernandez-Diaz} [J. Aust. Math. Soc., Ser. A 69, 41--60 (2000; Zbl 0990.47018)] and {\it A. Montes-Rodríguez} [J. Lond. Math. Soc., II. Ser. 61, 872--884 (2000; Zbl 0959.47016)].

47B33Composition operators
30D45Bloch functions, normal functions, normal families
30H05Bounded analytic functions
Full Text: DOI Link
[1] J. Arazy, Multipliers of Bloch functions , University of Haifa Publication Series 54 , 1982.
[2] L. Brown and A.L. Shields, Multipliers and cyclic vectors in the Bloch space , Michigan Math. J. 38 (1991), 141-146. · Zbl 0749.30023 · doi:10.1307/mmj/1029004269
[3] P.L. Duren, Theory of $H^p$ spaces , Academic Press, New York, 1970. · Zbl 0215.20203
[4] G.H. Hardy and J.E. Littlewood, Some properties of fractional integrals II, Math. Z. 34 (1932), 403-439. · Zbl 0003.15601 · doi:10.1007/BF01180596 · eudml:168318
[5] K.M. Madigan, Composition operators on analytic Lipschitz spaces , Proc. Amer. Math. Soc. 119 (1993), 465-473. JSTOR: · Zbl 0793.47037 · doi:10.2307/2159930 · http://links.jstor.org/sici?sici=0002-9939%28199310%29119%3A2%3C465%3ACOOALS%3E2.0.CO%3B2-3&origin=euclid
[6] K.M. Madigan and A. Matheson, Compact composition operators on the Bloch space , Trans. Amer. Math. Soc. 347 (1995), 2679-2687. JSTOR: · Zbl 0826.47023 · doi:10.2307/2154848 · http://links.jstor.org/sici?sici=0002-9947%28199507%29347%3A7%3C2679%3ACCOOTB%3E2.0.CO%3B2-W&origin=euclid
[7] S. Ohno and R. Zhao, Weighted composition operators on the Bloch space , Bull. Austral. Math. Soc., 63 (2001), 177-185. · Zbl 0985.47022 · doi:10.1017/S0004972700019250
[8] R.C. Roan, Composition operators on a space of Lipschitz functions , Rocky Mountain J. Math. 10 (1980), 371-379. · Zbl 0433.46023 · doi:10.1216/RMJ-1980-10-2-371
[9] J.H. Shapiro, Composition operators and classical function theory , Springer-Verlag, New York, 1993. · Zbl 0791.30033
[10] J. Xiao, Composition operators associated with Block-type spaces , Complex Variables Theory Appl. 46 (2001), 109-121. · Zbl 1044.47020
[11] K.H. Zhu, Operator theory on function spaces , Marcel Dekker, New York, 1990. · Zbl 0706.47019
[12] --------, Bloch type spaces of analytic functions , Rocky Mountain J. Math. 23 (1993), 1143-1177. · Zbl 0787.30019 · doi:10.1216/rmjm/1181072549