UR Scholarship Repository

Title

Bergman Spaces on Disconnected Domains

Authors

Alexandru Aleman
Stefan Richter
William T. Ross, University of RichmondFollow

Document Type

Technical Report

Publication Date

7-6-1994

Abstract

For a bounded region G ⊂ ℂ and a compact set K ⊂G , with area measure zero, we will characterize the invariant subspaces M (under ƒ → z ƒ) of the Bergman space L^p_a(G\K), 1 ≤ p < ∞, which contain L<sup>p_a(G) and with dim(M/(z-⋋)M) = 1 for all ⋋ ∈ G\K. When G\K is connected, we will see that dim(M/(z-⋋)M) = 1 for all ⋋ ∈ G\K and this in this case we will have a complete description of the invariant subspaces lying between L<sup>p_a(G) and L^p_a(G\K). When G\K is not connected, we will show that in general the invariant subspaces between L^p_a(G) and L^p_a(G\K) are fantastically complicated. As an application of these results, we will remark on the complexity on the invariant subspaces (under ƒ → ζ ƒ) of certain Besov spaces on K. In particular, we shall see that in the harmonic Dirichlet space B¹₂(∏), there are invariant subspaces F such that the dimension of ζF in F is infinite.

Comments

Copyright © 1994, Alexandru Aleman, Stefan Richter, and William T. Ross, University of Richmond, Richmond, Virginia.

Please note that downloads of the technical report are for private/personal use only.

Recommended Citation

Alexandru Aleman, Stefan Richter, and William T. Ross. Bergman Spaces on Disconnected Domains. Technical paper (TR-94-04). Math and Computer Science Technical Report Series. Richmond, Virginia: Department of Mathematics and Computer Science, University of Richmond, July 6, 1994.