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Invariant subspaces of the harmonic Dirichlet space with large co-dimension

Author: William T. Ross
Journal: Proc. Amer. Math. Soc. 124 (1996), 1841-1846
MSC (1991): Primary 30H05; Secondary 30C15
MathSciNet review: 1307561
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Abstract: In this paper, we comment on the complexity of the invariant subspaces (under the bilateral Dirichlet shift $f \to \zeta f$) of the harmonic Dirichlet space $D$. Using the sampling theory of Seip and some work on invariant subspaces of Bergman spaces, we will give examples of invariant subspaces ${\mathcal F} \subset D$ with $\mbox {dim}({\mathcal F}/ \zeta {\mathcal F}) = n$, $n \in % {\mathbb N} \cup \{\infty \}$. We will also generalize this to the Dirichlet classes $D_{\alpha }$, $0 < \alpha < \infty $, as well as the Besov classes $B^{\alpha }_{p}$, $1 < p < \infty $, $0 < \alpha < 1$.

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Additional Information

William T. Ross
Affiliation: Department of Mathematics University of Richmond Richmond, Virginia 23173

Keywords: Dirichlet spaces, invariant subspaces, co-dimension, Bergman spaces
Received by editor(s): October 31, 1994
Received by editor(s) in revised form: December 9, 1994
Additional Notes: This research was supported in part by a grant from the National Science Foundation.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society