# Composition operators: ${N}_{\alpha}$ to the Bloch space to ${Q}_{\beta}$

Studia Mathematica (2000)

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

## Access Full Article

top## Abstract

top## How to cite

topXiao, 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- [AS] A. Aleman and A. G. Siskakis, An integral operator on ${H}^{p}$, Complex Variables 28 (1995), 149-158.
- [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
- [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.
- [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
- [AXZ] R. Aulaskari, J. Xiao and R. Zhao, On subspaces and subclasses of BMOA and UBC, Analysis 15 (1995), 101-121. Zbl0835.30027
- [Ba] A. Baernstein II, Analytic functions of bounded mean oscillation, in: Aspects of Contemporary Complex Analysis, Academic Press, London, 1980, 2-26.
- [BCM] P. S. Bourdon, J. A. Cima and A. L. Matheson, Compact composition operators on BMOA, Trans. Amer. Math. Soc. 351 (1999), 2183-2196.
- [EX] M. Essén and J. Xiao, Some results on ${Q}_{p}$ spaces, 0 < p < 1, J. Reine Angew. Math. 485 (1997), 173-195.
- [J] H. Jarchow, Locally Convex Spaces, Teubner, 1981. Zbl0466.46001
- [JX] H. Jarchow and J. Xiao, Composition operators between Nevanlinna classes and Bergman spaces with weights, J. Operator Theory, to appear. Zbl0996.47031
- [L] D. H. Luecking, Trace ideal criteria for Toeplitz operators, J. Funct. Anal. 73 (1987), 345-368. Zbl0618.47018
- [MM] K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), 2679-2687. Zbl0826.47023
- [NX] A. Nicolau and J. Xiao, Bounded functions in Möbius invariant Dirichlet spaces, J. Funct. Anal. 150 (1997), 383-425. Zbl0886.30036
- [RU] W. Ramey and D. Ullrich, Bounded mean oscillation of Bloch pull-backs, Math. Ann. 291 (1991), 591-606. Zbl0727.32002
- [SS] J. H. Shapiro and A. L. Shields, Unusual topological properties of the Nevanlinna Class, Amer. J. Math. 97 (1976), 915-936. Zbl0323.30033
- [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.
- [SZ] W. Smith and R. Zhao, Composition operators mapping into the ${Q}_{p}$ spaces, Analysis 17 (1997), 239-263. Zbl0901.47023
- [Str] K. Stroethoff, Nevanlinna-type characterizations for the Bloch space and related spaces, Proc. Edinburgh Math. Soc. 33 (1990), 123-142. Zbl0703.30032
- [T] M. Tjani, Compact composition operators on some Möbius invariant Banach spaces, Ph.D. Thesis, Michigan State Univ. 1996. Zbl0963.47023
- [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
- [X2] J. Xiao, Compact composition operators on the area-Nevanlinna class, Exposition. Math. 17 (1999), 255-264. Zbl0977.30031
- [Z] K. Zhu, Operator Theory in Function Spaces, Dekker, New York, 1990. Zbl0706.47019

## NotesEmbed ?

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