- Open Access
Weighted composition followed and proceeded by differentiation operators fromZygmund spaces to Bloch-type spaces
© Long et al.; licensee Springer. 2014
Received: 12 January 2014
Accepted: 8 April 2014
Published: 2 May 2014
The boundedness and compactness of the weighted composition followed andproceeded by differentiation operators from Zygmund spaces to Bloch-type spacesand little Bloch-type spaces are characterized.
MSC: 47B38, 30H30, 32C15.
This space is called the little α-Bloch space (see ).
Then μ is called a normal function (see ).
It is well known that is a Banach space with the norm (see ).
This space is called the little Bloch-type space. When , the induced space becomes the α-Bloch space.
They are called weighted composition followed and proceeded by differentiationoperators, respectively, where and D are composition and differentiationoperators respectively. Associated with φ is the composition operator and weighted composition operator for and . It is interesting to provide a function theoreticcharacterization for φ inducing a bounded or compact compositionoperator, weighted composition operator and related ones on various spaces (see,e.g., [10–19]). For example, it is well known that is bounded on the classical Hardy, Bloch and Bergmanspaces. Operators and as well as some other products of linear operatorswere studied, for example, in [20–29] (see also the references therein). There has been some considerablerecent interest in investigation various type of operators from or to Zygmund typespaces (see, [7, 11, 23, 30–37]).
In this paper, we investigate the operators and from Zygmund spaces to Bloch-type spaces and littleBloch-type spaces. Some sufficient and necessary conditions for the boundedness andcompactness of these operators are given.
Throughout this paper, constants are denoted by C, they are positive and maydiffer from one occurrence to the other. The notation means that there is a positive constant Csuch that .
2 Main results and proofs
In this section, we state and prove our main results. In order to formulate our mainresults, we quote several lemmas which will be used in the proofs of the mainresults in this paper. The following lemma can be proved in a standard way (see,e.g., Proposition 3.11 in ). Hence we omit the details.
Lemma 2.1 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Thenis compact if and only ifis bounded and for any bounded sequencein (or) which converges to zero uniformly on compactsubsets of Δ as, and (or) as.
- (ii)⇒ (iii). Assume that is bounded, i.e., there exists a constant C such that
- (iii)⇒ (i). Assume that (2.1) and (2.2) hold. Then, for every , from (1.4), we have(2.15)
Taking the supremum in (2.15) for , and employing (2.1) and (2.2), we deduce that is bounded. The proof of Theorem 2.3 iscompleted. □
- (iii)is bounded,(2.16)
- (ii)⇒ (iii). Assume that is compact. Then it is clear that is bounded. By Theorem 2.3 we know that is bounded. Let be a sequence in Δ such that as and , (if such a sequence does not exist then (2.16) and (2.17) are vacuously satisfied). Set(2.18)
if one of these two limits exists.
- (iii)⇒ (i). Suppose that is bounded and conditions (2.16) and (2.17) hold. From Theorem 2.3, it follows that(2.22)
By combining this with Lemma 2.1 the result easily follows. The proof ofTheorem 2.4 is completed. □
Proof of Theorem 2.5 Assume that is bounded. Then, it is clear that is bounded. Taking the test functions and respectively, we obtain (2.25).
as . Since the operator is bounded, we have , which implies the boundedness of. □
- (ii)⇒ (iii). Assume that is compact. Then is bounded. From the proof of Theorem 2.5, we have(2.29)
from which the result follows in this case.
- (iii)⇒ (i). Let . Then we have
From Lemma 2.2 it follows that the operator is compact. □
Similarly to the proofs of Theorems 2.3-2.6, we can get the following results; weomit the proof.
- (iii)is bounded,
The authors would like to thank the referees for their valuable suggestions whichgreatly improve the present article. The work was supported by the UnitedTechnology Foundation of Science and Technology Department of Guizhou Provinceand Guizhou Normal University (Grant No. LKS12), and National NaturalScience Foundation of China (Grant No. 11171080, Grant No. 11171277).
- Zhu K: Bloch type spaces of analytic functions. Rocky Mt. J. Math. 1993,23(3):1143–1177. 10.1216/rmjm/1181072549View ArticleMathSciNetMATHGoogle Scholar
- Shields A, Williams D: Bonded projections, duality, and multipliers in spaces of analyticfunctions. Trans. Am. Math. Soc. 1971, 162: 287–302.MathSciNetGoogle Scholar
- Fu X, Zhu X: Weighted composition operators on some weighted spaces in the unit ball. Abstr. Appl. Anal. 2008., 2008: Article ID 605807Google Scholar
- Hu Z, Wang S: Composition operators on Bloch-type spaces. Proc. R. Soc. Edinb. A 2005,135(6):1229–1239. 10.1017/S0308210500004340View ArticleMathSciNetMATHGoogle Scholar
- Krantz SG, Stević S: On the iterated logarithmic Bloch space on the unit ball. Nonlinear Anal. TMA 2009,71(5–6):1772–1795. 10.1016/j.na.2009.01.013View ArticleMathSciNetMATHGoogle Scholar
- Duren P: Theory of Hp Spaces. Academic Press, New York; 1970.MATHGoogle Scholar
- Li S, Stević S: Volterra type operators on Zygmund spaces. J. Inequal. Appl. 2007., 2007: Article ID 32124Google Scholar
- Stević S: On an integral operator from the Zygmund space to the Bloch-type space on theunit ball. Glasg. Math. J. 2009,51(2):275–287. 10.1017/S0017089508004692MathSciNetView ArticleMATHGoogle Scholar
- Stević S: On an integral-type operator from Zygmund-type spaces to mixed-norm spaces onthe unit ball. Abstr. Appl. Anal. 2010., 2010: Article ID 198608Google Scholar
- Cowen C, Maccluer B Studies in Advanced Mathematics. In Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton; 1995.Google Scholar
- Li S, Stević S: Generalized composition operators on Zygmund spaces and Bloch type spaces. J. Math. Anal. Appl. 2008,338(2):1282–1295. 10.1016/j.jmaa.2007.06.013MathSciNetView ArticleMATHGoogle Scholar
- Madigan K, Matheson A: Compact composition operators on the Bloch space. Trans. Am. Math. Soc. 1995, 347: 2679–2687. 10.1090/S0002-9947-1995-1273508-XMathSciNetView ArticleMATHGoogle Scholar
- Shapiro J: Composition Operators and Classical Function Theory. Springer, New York; 1993.View ArticleMATHGoogle Scholar
- Stević S: Generalized composition operators between mixed norm space and some weightedspaces. Numer. Funct. Anal. Optim. 2009, 29: 426–434.Google Scholar
- Stević S:Norm of weighted composition operators from Bloch space to on the unit ball. Ars Comb. 2008, 88: 125–127.MathSciNetMATHGoogle Scholar
- Stević S: Norm of weighted composition operators from α -Bloch spaces toweighted-type spaces. Appl. Math. Comput. 2009, 215: 818–820. 10.1016/j.amc.2009.06.005MathSciNetView ArticleMATHGoogle Scholar
- Ueki S:Composition operators on the Privalov spaces of the unit ball of. J. Korean Math. Soc. 2005,42(1):111–127.MathSciNetView ArticleMATHGoogle Scholar
- Ueki S: Weighted composition operators on the Bargmann-Fock space. Int. J. Mod. Math. 2008,3(3):231–243.MathSciNetMATHGoogle Scholar
- Zhu X: Generalized weighted composition operators from Bloch-type spaces to weightedBergman spaces. Indian J. Math. 2007,49(2):139–149.MathSciNetMATHGoogle Scholar
- Hibschweiler R, Portnoy N: Composition followed by differentiation between Bergman and Hardy spaces. Rocky Mt. J. Math. 2005,35(3):843–855. 10.1216/rmjm/1181069709MathSciNetView ArticleMATHGoogle Scholar
- Li S, Stević S: Composition followed by differentiation between Bloch type spaces. J. Comput. Anal. Appl. 2007,9(2):195–205.MathSciNetMATHGoogle Scholar
- Li S, Stević S: Composition followed by differentiation between and α -Bloch spaces. Houst. J. Math. 2009,35(1):327–340.MathSciNetMATHGoogle Scholar
- Li S, Stević S: Products of composition and differentiation operators from Zygmund spaces toBloch spaces and Bers spaces. Appl. Math. Comput. 2010, 217: 3144–3154. 10.1016/j.amc.2010.08.047MathSciNetView ArticleMATHGoogle Scholar
- Li S, Stević S: Composition followed by differentiation from mixed-norm spaces to α -Bloch spaces. Sb. Math. 2008,199(12):1847–1857. 10.1070/SM2008v199n12ABEH003983MathSciNetView ArticleMATHGoogle Scholar
- Long J, Wu P:Weighted composition followed and proceeded by differentiation operators from spaces to Bloch-type spaces. J. Inequal. Appl. 2012., 2012: Article ID 160Google Scholar
- Ohno S: Products of composition and differentiation between Hardy spaces. Bull. Aust. Math. Soc. 2006,73(2):235–243. 10.1017/S0004972700038818MathSciNetView ArticleMATHGoogle Scholar
- Stević S: Weighted differentiation composition operators from mixed-norm spaces toweighted-type spaces. Appl. Math. Comput. 2009, 211: 222–233. 10.1016/j.amc.2009.01.061MathSciNetView ArticleMATHGoogle Scholar
- Stević S: Norm and essential norm of composition followed by differentiation from α −Bloch spaces to . Appl. Math. Comput. 2009,207(1):225–229. 10.1016/j.amc.2008.10.032MathSciNetView ArticleMATHGoogle Scholar
- Stević S: Products of composition and differentiation operators on the weighted Bergmanspace. Bull. Belg. Math. Soc. Simon Stevin 2009,16(4):623–635.MathSciNetMATHGoogle Scholar
- Fu X, Li S: Composition operators from Zygmund spaces into QK spaces. J. Inequal. Appl. 2013., 2013: Article ID 175Google Scholar
- Li S, Stević S: On an integral-type operator from ω -Bloch spaces to μ -Zygmund spaces. Appl. Math. Comput. 2010,215(12):4385–4391. 10.1016/j.amc.2009.12.070MathSciNetView ArticleMATHGoogle Scholar
- Li S, Stević S: Weighted composition operators from Zygmund spaces into Bloch spaces. Appl. Math. Comput. 2008,206(2):825–831. 10.1016/j.amc.2008.10.006MathSciNetView ArticleMATHGoogle Scholar
- Li S, Stević S: Integral-type operators from Bloch-type spaces to Zygmund-type spaces. Appl. Math. Comput. 2009, 215: 464–473. 10.1016/j.amc.2009.05.011MathSciNetView ArticleMATHGoogle Scholar
- Li S, Stević S:Products of Volterra type operator and composition operator from and Bloch spaces to the Zygmund space. J. Math. Anal. Appl. 2008, 345: 40–52. 10.1016/j.jmaa.2008.03.063MathSciNetView ArticleMATHGoogle Scholar
- Liu Y, Yu Y: Riemann-Stieltjes operator from mixed norm spaces to Zygmund-type spaces onthe unit ball. Taiwan. J. Math. 2013,17(5):1751–1764.MATHGoogle Scholar
- Yang C:Integral-type operators from spaces to Zygmund-type spaces on the unitball. J. Inequal. Appl. 2010., 2010: Article ID 789285Google Scholar
- Zhu X: Extended Cesàro operators from mixed norm spaces to Zygmund typespaces. Tamsui Oxford Univ. J. Math. Sci. 2010,26(4):411–422.MATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permitsunrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly credited.