New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space

Flavia Colonna

Open Mathematics (2013)

  • Volume: 11, Issue: 1, page 55-73
  • ISSN: 2391-5455

Abstract

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Let ψ and φ be analytic functions on the open unit disk 𝔻 with φ( 𝔻 ) ⊆ 𝔻 . We give new characterizations of the bounded and compact weighted composition operators W ψ,ϕ from the Hardy spaces H p, 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A αp, α > − 1,1 ≤ p < ∞, and the Dirichlet space 𝒟 to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W ψ,ϕ f for suitable collections of functions f in the respective spaces. We also obtain characterizations of boundedness for H 1 as well as of compactness for H p, 1 ≤ p < ∞, and 𝒟 purely in terms of the symbols ψ and φ.

How to cite

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Flavia Colonna. "New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space." Open Mathematics 11.1 (2013): 55-73. <http://eudml.org/doc/269568>.

@article{FlaviaColonna2013,
abstract = {Let ψ and φ be analytic functions on the open unit disk $\mathbb \{D\}$ with φ($\mathbb \{D\}$) ⊆ $\mathbb \{D\}$. We give new characterizations of the bounded and compact weighted composition operators W ψ,ϕ from the Hardy spaces H p, 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A αp, α > − 1,1 ≤ p < ∞, and the Dirichlet space $\mathcal \{D\}$ to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W ψ,ϕ f for suitable collections of functions f in the respective spaces. We also obtain characterizations of boundedness for H 1 as well as of compactness for H p, 1 ≤ p < ∞, and $\mathcal \{D\}$ purely in terms of the symbols ψ and φ.},
author = {Flavia Colonna},
journal = {Open Mathematics},
keywords = {Weighted composition operators; Bloch space; Hardy space; Weighted Bergman space; Dirichlet space; weighted composition operators; weighted Bergman space},
language = {eng},
number = {1},
pages = {55-73},
title = {New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space},
url = {http://eudml.org/doc/269568},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Flavia Colonna
TI - New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space
JO - Open Mathematics
PY - 2013
VL - 11
IS - 1
SP - 55
EP - 73
AB - Let ψ and φ be analytic functions on the open unit disk $\mathbb {D}$ with φ($\mathbb {D}$) ⊆ $\mathbb {D}$. We give new characterizations of the bounded and compact weighted composition operators W ψ,ϕ from the Hardy spaces H p, 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A αp, α > − 1,1 ≤ p < ∞, and the Dirichlet space $\mathcal {D}$ to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W ψ,ϕ f for suitable collections of functions f in the respective spaces. We also obtain characterizations of boundedness for H 1 as well as of compactness for H p, 1 ≤ p < ∞, and $\mathcal {D}$ purely in terms of the symbols ψ and φ.
LA - eng
KW - Weighted composition operators; Bloch space; Hardy space; Weighted Bergman space; Dirichlet space; weighted composition operators; weighted Bergman space
UR - http://eudml.org/doc/269568
ER -

References

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  1. [1] Anderson J.M., Clunie J., Pommerenke Ch., On Bloch functions and normal functions, J. Reine Angew. Math., 1974, 270, 12–37 Zbl0292.30030
  2. [2] Arazy J., Fisher S.D., The uniqueness of the Dirichlet space among Möbius-invariant Hilbert spaces, Illinois J. Math., 1985, 29(3), 449–462 Zbl0555.30021
  3. [3] Djrbashian A.E., Shamoian F.A., Topics in the Theory of A αp Spaces, Teubner-Texte Math., 105, Teubner, Leipzig, 1988 
  4. [4] Duren P.L., Theory of H p Spaces, Pure Appl. Math., 38, Academic Press, New York-London, 1970 Zbl0215.20203
  5. [5] Duren P., Schuster A., Bergman Spaces, Math. Surveys Monogr., 100, American Mathematical Society, Providence, 2004 
  6. [6] Hedenmalm H., Korenblum B., Zhu K., Theory of Bergman Spaces, Grad. Texts in Math., 199, Springer, New York, 2000 http://dx.doi.org/10.1007/978-1-4612-0497-8 Zbl0955.32003
  7. [7] Li S., Stevic S., Weighted composition operators from Bergman-type spaces into Bloch spaces, Proc. Indian Acad. Sci. Math. Sci., 2007, 117(3), 371–385 http://dx.doi.org/10.1007/s12044-007-0032-y Zbl1130.47016
  8. [8] Ohno S., Weighted composition operators between H 1 and the Bloch space, Taiwanese J. Math., 2001, 5(3), 555–563 Zbl0997.47025
  9. [9] Ohno S., Stroethoff K., Weighted composition operators from reproducing Hilbert spaces to Bloch spaces, Houston J. Math., 2011, 37(2), 537–558 Zbl1244.47029
  10. [10] Ohno S., Zhao R., Weighted composition operators on the Bloch space, Bull. Austral. Math. Soc., 2001, 63(2), 177–185 http://dx.doi.org/10.1017/S0004972700019250 Zbl0985.47022
  11. [11] Ross W.T., The classical Dirichlet space, In: Recent Advances in Operator-Related Function Theory, Dublin, August 4–6, 2004, Contemp. Math., 393, American Mathematical Society, Providence, 2006, 171–197 http://dx.doi.org/10.1090/conm/393/07378 
  12. [12] Sharma A.K., Kumari R., Weighted composition operators between Bergman and Bloch spaces, Commun. Korean Math. Soc., 2007, 22(3), 373–382 http://dx.doi.org/10.4134/CKMS.2007.22.3.373 Zbl1168.31301
  13. [13] Tjani M., Compact Composition Operators on Some Möbius Invariant Banach Spaces, PhD thesis, Michigan State University, 1996 
  14. [14] Tjani M., Compact composition operators on Besov spaces, Trans. Amer. Math. Soc., 2003, 355(11), 4683–4698 http://dx.doi.org/10.1090/S0002-9947-03-03354-3 Zbl1045.47020
  15. [15] Wulan H., Zheng D., Zhu K., Compact composition operators on BMOA and the Bloch space, Proc. Amer. Math. Soc., 2009, 137(11), 3861–3868 http://dx.doi.org/10.1090/S0002-9939-09-09961-4 Zbl1194.47038
  16. [16] Zhu K.H., Operator Theory in Function Spaces, Monogr. Textbooks Pure Appl. Math., 139, Marcel Dekker, New York, 1990 Zbl0706.47019

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