Bounded composition operators with closed range on the Dirichlet space
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- by Daniel H. Luecking PDF
- Proc. Amer. Math. Soc. 128 (2000), 1109-1116 Request permission
Abstract:
For composition operators on spaces of analytic functions it is well known that norm estimates can be converted to Carleson measure estimates. The boundedness of the composition operator becomes equivalent to a Carleson measure inequality. The measure corresponding to a composition operator $C_\varphi$ on the Dirichet space $\mathcal D$ is $d\nu _\varphi = n_\varphi dA$, where $n_\varphi (z)$ is the cardinality of the preimage $\varphi ^{-1}(z)$. The composition operator will have closed range if and only if the corresponding measure satisfies a “reverse Carleson measure” theorem: $\| f \|_{\mathcal {D}}^2 \le \int |f’|^2 d\nu _\varphi$ for all $f\in \mathcal D$. Assuming $C_\varphi$ is bounded, a necessary condition for this inequality is a reverse of the Carleson condition: (C) $\nu _\varphi (S) \ge c |S|$ for all Carleson squares $S$. It has long been known that this is not sufficient for a completely general measure. Here we show that it is also not sufficient for the special measures $\nu _\varphi$. That is, we construct a function $\varphi$ such that $C_\varphi$ is bounded and $\nu _\varphi$ satisfies (C) but the composition operator $C_\varphi$ does not have closed range.References
- Éric Amar, Suites d’interpolation pour les classes de Bergman de la boule et du polydisque de $\textbf {C}^{n}$, Canadian J. Math. 30 (1978), no. 4, 711–737 (French). MR 499309, DOI 10.4153/CJM-1978-062-6
- Mirjana Jovović and Barbara MacCluer, Composition operators on Dirichlet spaces, Acta Sci. Math. (Szeged) 63 (1997), no. 1-2, 229–247. MR 1459789
- Daniel H. Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives, Amer. J. Math. 107 (1985), no. 1, 85–111. MR 778090, DOI 10.2307/2374458
- Daniel H. Luecking, Dominating measures for spaces of analytic functions, Illinois J. Math. 32 (1988), no. 1, 23–39. MR 921348
- Daniel H. Luecking, Zero sequences for Bergman spaces, Complex Variables Theory Appl. 30 (1996), no. 4, 345–362. MR 1413164, DOI 10.1080/17476939608814936
- Richard Rochberg, Interpolation by functions in Bergman spaces, Michigan Math. J. 29 (1982), no. 2, 229–236. MR 654483
- Nina Zorboska, Composition operators on weighted Dirichlet spaces, Proc. Amer. Math. Soc. 126 (1998), no. 7, 2013–2023. MR 1443862, DOI 10.1090/S0002-9939-98-04266-X
Additional Information
- Daniel H. Luecking
- Affiliation: Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701
- Email: luecking@comp.uark.edu
- Received by editor(s): February 23, 1998
- Received by editor(s) in revised form: June 1, 1998
- Published electronically: August 17, 1999
- Communicated by: Albert Baernstein II
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1109-1116
- MSC (1991): Primary 46E20
- DOI: https://doi.org/10.1090/S0002-9939-99-05103-5
- MathSciNet review: 1637392