Composition operators: N α to the Bloch space to Q β

Jie Xiao

Studia Mathematica (2000)

  • Volume: 139, Issue: 3, page 245-260
  • ISSN: 0039-3223

Abstract

top
Let N α ,B and Qβ be the weighted Nevanlinna space, the Bloch space and the Q space, respectively. Note that B and Q β are Möbius invariant, but N α is not. We characterize, in function-theoretic terms, when the composition operator C ϕ f = f ◦ ϕ induced by an analytic self-map ϕ of the unit disk defines an operator C ϕ : N α → B , B → Q β , N α → Q β which is bounded resp. compact.

How to cite

top

Xiao, Jie. "Composition operators: $N_α$ to the Bloch space to $Q_β$." Studia Mathematica 139.3 (2000): 245-260. <http://eudml.org/doc/216721>.

@article{Xiao2000,
abstract = {Let $N_α$,B and Qβ be the weighted Nevanlinna space, the Bloch space and the Q space, respectively. Note that B and $Q_β$ are Möbius invariant, but $N_α$ is not. We characterize, in function-theoretic terms, when the composition operator $C_ϕ f=f◦ϕ$ induced by an analytic self-map ϕ of the unit disk defines an operator $C_ϕ:N_α→B$, $B→Q_β$, $N_α→Q_β$ which is bounded resp. compact.},
author = {Xiao, Jie},
journal = {Studia Mathematica},
keywords = {composition operator; boundedness; compactness; $N_α$; β; Q\_β; weighted Nevanlinna space; Bloch space; space; Bloch function; Carleson box; Möbius invariant version of the generalized Nevanlinna counting function},
language = {eng},
number = {3},
pages = {245-260},
title = {Composition operators: $N_α$ to the Bloch space to $Q_β$},
url = {http://eudml.org/doc/216721},
volume = {139},
year = {2000},
}

TY - JOUR
AU - Xiao, Jie
TI - Composition operators: $N_α$ to the Bloch space to $Q_β$
JO - Studia Mathematica
PY - 2000
VL - 139
IS - 3
SP - 245
EP - 260
AB - Let $N_α$,B and Qβ be the weighted Nevanlinna space, the Bloch space and the Q space, respectively. Note that B and $Q_β$ are Möbius invariant, but $N_α$ is not. We characterize, in function-theoretic terms, when the composition operator $C_ϕ f=f◦ϕ$ induced by an analytic self-map ϕ of the unit disk defines an operator $C_ϕ:N_α→B$, $B→Q_β$, $N_α→Q_β$ which is bounded resp. compact.
LA - eng
KW - composition operator; boundedness; compactness; $N_α$; β; Q_β; weighted Nevanlinna space; Bloch space; space; Bloch function; Carleson box; Möbius invariant version of the generalized Nevanlinna counting function
UR - http://eudml.org/doc/216721
ER -

References

top
  1. [AS] A. Aleman and A. G. Siskakis, An integral operator on H p , Complex Variables 28 (1995), 149-158. 
  2. [AL] R. Aulaskari and P. Lappan, Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal, in: Complex Analysis and its Applications, Pitman Res. Notes in Math. 305, Longman, 1994, 136-146. Zbl0826.30027
  3. [ANZ] R. Aulaskari, M. Norwak and R. Zhao, The n-the derivative characterizations of Möbius invariant Dirichlet spaces, Bull. Austral. Math. Soc. 58 (1998), 43-56. 
  4. [ASX] R. Aulaskari, D. Stegenga and J. Xiao, Some subclasses of BMOA and their characterization in terms of Carleson measures, Rocky Mountain J. Math. 26 (1996), 485-506. Zbl0861.30033
  5. [AXZ] R. Aulaskari, J. Xiao and R. Zhao, On subspaces and subclasses of BMOA and UBC, Analysis 15 (1995), 101-121. Zbl0835.30027
  6. [Ba] A. Baernstein II, Analytic functions of bounded mean oscillation, in: Aspects of Contemporary Complex Analysis, Academic Press, London, 1980, 2-26. 
  7. [BCM] P. S. Bourdon, J. A. Cima and A. L. Matheson, Compact composition operators on BMOA, Trans. Amer. Math. Soc. 351 (1999), 2183-2196. 
  8. [EX] M. Essén and J. Xiao, Some results on Q p spaces, 0 < p < 1, J. Reine Angew. Math. 485 (1997), 173-195. 
  9. [J] H. Jarchow, Locally Convex Spaces, Teubner, 1981. Zbl0466.46001
  10. [JX] H. Jarchow and J. Xiao, Composition operators between Nevanlinna classes and Bergman spaces with weights, J. Operator Theory, to appear. Zbl0996.47031
  11. [L] D. H. Luecking, Trace ideal criteria for Toeplitz operators, J. Funct. Anal. 73 (1987), 345-368. Zbl0618.47018
  12. [MM] K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), 2679-2687. Zbl0826.47023
  13. [NX] A. Nicolau and J. Xiao, Bounded functions in Möbius invariant Dirichlet spaces, J. Funct. Anal. 150 (1997), 383-425. Zbl0886.30036
  14. [RU] W. Ramey and D. Ullrich, Bounded mean oscillation of Bloch pull-backs, Math. Ann. 291 (1991), 591-606. Zbl0727.32002
  15. [SS] J. H. Shapiro and A. L. Shields, Unusual topological properties of the Nevanlinna Class, Amer. J. Math. 97 (1976), 915-936. Zbl0323.30033
  16. [ST] J. H. Shapiro and P. D. Taylor, Compact, nuclear, and Hilbert-Schmidt composition operators on H 2 , Indiana Univ. Math. J. 23 (1973), 471-496. 
  17. [SZ] W. Smith and R. Zhao, Composition operators mapping into the Q p spaces, Analysis 17 (1997), 239-263. Zbl0901.47023
  18. [Str] K. Stroethoff, Nevanlinna-type characterizations for the Bloch space and related spaces, Proc. Edinburgh Math. Soc. 33 (1990), 123-142. Zbl0703.30032
  19. [T] M. Tjani, Compact composition operators on some Möbius invariant Banach spaces, Ph.D. Thesis, Michigan State Univ. 1996. Zbl0963.47023
  20. [X1] J. Xiao, Carleson measure, atomic decomposition and free interpolation from Bloch space, Ann. Acad. Sci. Fenn. Ser. A.I. Math. 19 (1994), 35-44. Zbl0816.30025
  21. [X2] J. Xiao, Compact composition operators on the area-Nevanlinna class, Exposition. Math. 17 (1999), 255-264. Zbl0977.30031
  22. [Z] K. Zhu, Operator Theory in Function Spaces, Dekker, New York, 1990. Zbl0706.47019

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.