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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Bounded composition operators
with closed range on the Dirichlet space

Author: Daniel H. Luecking
Journal: Proc. Amer. Math. Soc. 128 (2000), 1109-1116
MSC (1991): Primary 46E20
Published electronically: August 17, 1999
MathSciNet review: 1637392
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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.

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Additional Information

Daniel H. Luecking
Affiliation: Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701

Keywords: Composition operator, closed range
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
Article copyright: © Copyright 2000 American Mathematical Society