Composition operators: $N_α$ to the Bloch space to $Q_β$

Xiao, Jie

Composition operators:

N_{α}

to the Bloch space to

Q_{β}

Xiao, Jie

Studia Mathematica, Tome 141 (2000), p. 245-260 / Harvested from The Polish Digital Mathematics Library

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Résumé

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 $C_{ϕ} : N_{α} \to B$ , $B \to Q_{β}$ , $N_{α} \to Q_{β}$ which is bounded resp. compact.

Publié le : 2000-01-01

Zbl 0963.30021

EUDML-ID : urn:eudml:doc:216721

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     title = {Composition operators: $N\_$\alpha$$ to the Bloch space to $Q\_$\beta$$
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     journal = {Studia Mathematica},
     volume = {141},
     year = {2000},
     pages = {245-260},
     zbl = {0963.30021},
     language = {en},
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Xiao, Jie. Composition operators: $N_α$ to the Bloch space to $Q_β$
            . Studia Mathematica, Tome 141 (2000) pp. 245-260. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv139i3p245bwm/

Bibliographie

[00000] [AS] A. Aleman and A. G. Siskakis, An integral operator on $H^{p}$ , Complex Variables 28 (1995), 149-158.

[00001] [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. | Zbl 0826.30027

[00002] [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.

[00003] [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. | Zbl 0861.30033

[00004] [AXZ] R. Aulaskari, J. Xiao and R. Zhao, On subspaces and subclasses of BMOA and UBC, Analysis 15 (1995), 101-121. | Zbl 0835.30027

[00005] [Ba] A. Baernstein II, Analytic functions of bounded mean oscillation, in: Aspects of Contemporary Complex Analysis, Academic Press, London, 1980, 2-26.

[00006] [BCM] P. S. Bourdon, J. A. Cima and A. L. Matheson, Compact composition operators on BMOA, Trans. Amer. Math. Soc. 351 (1999), 2183-2196.

[00007] [EX] M. Essén and J. Xiao, Some results on $Q_{p}$ spaces, 0 < p < 1, J. Reine Angew. Math. 485 (1997), 173-195.

[00008] [J] H. Jarchow, Locally Convex Spaces, Teubner, 1981. | Zbl 0466.46001

[00009] [JX] H. Jarchow and J. Xiao, Composition operators between Nevanlinna classes and Bergman spaces with weights, J. Operator Theory, to appear. | Zbl 0996.47031

[00010] [L] D. H. Luecking, Trace ideal criteria for Toeplitz operators, J. Funct. Anal. 73 (1987), 345-368. | Zbl 0618.47018

[00011] [MM] K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), 2679-2687. | Zbl 0826.47023

[00012] [NX] A. Nicolau and J. Xiao, Bounded functions in Möbius invariant Dirichlet spaces, J. Funct. Anal. 150 (1997), 383-425. | Zbl 0886.30036

[00013] [RU] W. Ramey and D. Ullrich, Bounded mean oscillation of Bloch pull-backs, Math. Ann. 291 (1991), 591-606. | Zbl 0727.32002

[00014] [SS] J. H. Shapiro and A. L. Shields, Unusual topological properties of the Nevanlinna Class, Amer. J. Math. 97 (1976), 915-936. | Zbl 0323.30033

[00015] [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.

[00016] [SZ] W. Smith and R. Zhao, Composition operators mapping into the $Q_{p}$ spaces, Analysis 17 (1997), 239-263. | Zbl 0901.47023

[00017] [Str] K. Stroethoff, Nevanlinna-type characterizations for the Bloch space and related spaces, Proc. Edinburgh Math. Soc. 33 (1990), 123-142. | Zbl 0703.30032

[00018] [T] M. Tjani, Compact composition operators on some Möbius invariant Banach spaces, Ph.D. Thesis, Michigan State Univ. 1996. | Zbl 0963.47023

[00019] [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. | Zbl 0816.30025

[00020] [X2] J. Xiao, Compact composition operators on the area-Nevanlinna class, Exposition. Math. 17 (1999), 255-264. | Zbl 0977.30031

[00021] [Z] K. Zhu, Operator Theory in Function Spaces, Dekker, New York, 1990. | Zbl 0706.47019