Composition operators on Dirichlet-type spaces

Author:
R. A. Hibschweiler

Journal:
Proc. Amer. Math. Soc. **128** (2000), 3579-3586

MSC (2000):
Primary 47B38; Secondary 30H05

DOI:
https://doi.org/10.1090/S0002-9939-00-05886-X

Published electronically:
August 17, 2000

MathSciNet review:
1778280

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

The Dirichlet-type space ) is the Banach space of functions analytic in the unit disc with derivatives belonging to the Bergman space . Let be an analytic self-map of the disc and define for . The operator is bounded (respectively, compact) if and only if a related measure is Carleson (respectively, compact Carleson). If is bounded (or compact) on , then the same behavior holds on ) and on the weighted Dirichlet space . Compactness on implies that is compact on the Hardy spaces and the angular derivative exists nowhere on the unit circle. Conditions are given which, together with the angular derivative condition, imply compactness on the space . Inner functions which induce bounded composition operators on are discussed briefly.

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Additional Information

**R. A. Hibschweiler**

Affiliation:
Department of Mathematics, University of New Hampshire, Durham, New Hampshire 03824

Email:
rah2@cisunix.unh.edu

DOI:
https://doi.org/10.1090/S0002-9939-00-05886-X

Keywords:
Composition operator,
Dirichlet space,
Carleson measure,
angular derivative

Received by editor(s):
October 16, 1998

Received by editor(s) in revised form:
February 12, 1999

Published electronically:
August 17, 2000

Communicated by:
David R. Larson

Article copyright:
© Copyright 2000
American Mathematical Society