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)
Products of Volterra type operator and composition operator from H and Bloch spaces to Zygmund spaces. (English) Zbl 1145.47022
The Zygmund space 𝒵 is the set of all analytic functions f on the unit disc 𝔻 such that f 𝒵 =|f(0)|+|f ' (0)|+sup z (1-|z| 2 )|f '' (z)|<+, endowed with such a norm. If lim |z|1 (1-|z| 2 )|f '' (z)|=0, then f is said to belong to the little Zygmund space 𝒵 0 · Given an analytic function gH(𝔻), two types of Volterra integral operator are defined according to J g (f)(z)= 0 z fg ' and I g (f)(z)= 0 z f ' g for fH(𝔻)· The authors consider 𝒵 and 𝒵 0 valued compositions of these operators and composition operators C φ whose symbol φ is an analytic self-map of 𝔻· They compare their boundedness or compactness regarding the domain space, specifically, H , the Bloch space , or the little Bloch space 0 · As a sample of the paper results, let us quote the following (Theorem 1): Set T=C φ I g · Then T:H 𝒵 is bounded if and only if T:𝒵 is bounded if and only if T: 0 𝒵 is bounded. Similarly, for the compact case.

MSC:
47B33Composition operators
47B38Operators on function spaces (general)
References:
[1]Benke, G.; Chang, D. C.: A note on weighted Bergman spaces and the Cesàro operator, Nagoya math. J. 159, 25-43 (2000) · Zbl 0981.32001
[2]Chang, D. C.; Li, S.; Stević, S.: On some integral operators on the unit polydisk and the unit ball, Taiwanese J. Math. 11, No. 5, 1251-1286 (2007) · Zbl 1149.47026
[3]Chang, D. C.; Stević, S.: Estimates of an integral operator on function spaces, Taiwanese J. Math. 7, No. 3, 423-432 (2003) · Zbl 1052.47044
[4]Cowen, C. C.; Maccluer, B. D.: Composition operators on spaces of analytic functions, (1995) · Zbl 0873.47017
[5]Dunford, N.; Schwartz, J. T.: Linear operators I, (1958) · Zbl 0084.10402
[6]Duren, P. L.: Theory of hp spaces, (1970) · Zbl 0215.20203
[7]Hu, Z. J.: Extended Cesàro operators on mixed norm spaces, Proc. amer. Math. soc. 131, No. 7, 2171-2179 (2003) · Zbl 1054.47023 · doi:10.1090/S0002-9939-02-06777-1
[8]Hu, Z. J.: Extended Cesàro operators on the Bloch space in the unit ball of cn, Acta math. Sci. ser. B engl. Ed. 23, No. 4, 561-566 (2003) · Zbl 1044.47023
[9]Hu, Z. J.: Extended Cesàro operators on Bergman spaces, J. math. Anal. appl. 296, 435-454 (2004)
[10]Li, S.: Riemann – Stieltjes operators from F(p,q,s) to Bloch space on the unit ball, J. inequal. Appl. 2006 (2006) · Zbl 1131.47030 · doi:10.1155/JIA/2006/27874
[11]Li, S.; Stević, S.: Riemann – stieltjies type integral operators on the unit ball in cn, Complex var. Elliptic funct. 52, No. 6, 495-517 (2007) · Zbl 1124.47022 · doi:10.1080/17476930701235225
[12]Li, S.; Stević, S.: Volterra-type operators on Zygmund spaces, J. inequal. Appl. 2007 (2007) · Zbl 1146.30303 · doi:10.1155/2007/32124
[13]Madigan, K.; Matheson, A.: Compact composition operators on the Bloch space, Trans. amer. Math. soc. 347, No. 7, 2679-2687 (1995) · Zbl 0826.47023 · doi:10.2307/2154848
[14]Ohno, S.: Weighted composition operators between H and the Bloch space, Taiwanese J. Math. 5, No. 2, 555-563 (2001) · Zbl 0997.47025
[15]Pommerenke, Ch.: Schlichte funktionen und analytische funktionen von beschränkter mittlerer oszillation, Comment. math. Helv. 52, 591-602 (1977) · Zbl 0369.30012 · doi:10.1007/BF02567392
[16]Siskakis, A. G.; Zhao, R.: A Volterra type operator on spaces of analytic functions, Contemp. math. 232, 299-311 (1999) · Zbl 0955.47029
[17]Stević, S.: On an integral operator on the unit ball in cn, J. inequal. Appl. 1, 81-88 (2005) · Zbl 1074.47013 · doi:10.1155/JIA.2005.81
[18]Stević, S.: Boundedness and compactness of an integral operator on mixed norm spaces on the polydisc, Sibirsk. mat. Zh. 48, No. 3, 694-706 (2007) · Zbl 1164.47331 · doi:emis:journals/SMZ/2007/03/694.htm
[19]Tang, X.: Extended Cesàro operators between Bloch-type spaces in the unit ball of cn, J. math. Anal. appl. 326, 1199-1211 (2007) · Zbl 1117.47022 · doi:10.1016/j.jmaa.2006.03.082
[20]Zhu, K.: Bloch type spaces of analytic functions, Rocky mountain J. Math. 23, No. 3, 1143-1177 (1993) · Zbl 0787.30019 · doi:10.1216/rmjm/1181072549