#### DOI

10.4153/CJM-1996-011-5

#### Abstract

For a bounded region *G *C C and a compact set *K *C G, with area measure zero, we will characterize the invariant subspaces *M *(under *f* -> *zf)*of the Bergman space *L ^{p}_{a}(G \ K), *1 ≤

*p <*∞, which contain

*L*and with dim(

^{p}_{a}(G)*M*/(z - λ

*)M)*= 1 for all λϵ

*G \ K.*When

*G \ K*is connected, we will see that

*di\m(M /(z*— λ)

*M*) = 1 for all λ ϵ G \

*K*and thus in this case we will have a complete description of the invariant subspaces lying between

*L*and

^{p}_{a}(G)*L*When

^{p}_{a}(G \ K).*p*= ∞, we will remark on the structure of the weak-star closed z-invariant subspaces between

*H*and

^{∞}(G)*H*When G \

^{∞}(G \ K).*K*is not connected, we will show that in general the invariant subspaces between

*L*and

^{p}_{a}(G)*L*are fantastically complicated. As an application of these results, we will remark on the complexity of the invariant subspaces (under

^{p}_{a}(G \ K)*f-> Cf)*of certain Besov spaces on

*K.*In particular, we shall see that in the harmonic Dirichlet space B

^{1}

_{2}(T), there are invariant subspaces

*F*such that the dimension of (

*F*in

*F*is infinite.

#### Document Type

Article

#### Publication Date

1996

#### Publisher Statement

Copyright © 1996 Canadian Mathematical Society. This article first appeared in *Canadian Journal of Mathematics *48 (1996), 225-243.

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#### Recommended Citation

Aleman, Alexandru, Stefan Richter, and William T. Ross. "Bergman Spaces on Disconnected Domains." *Canadian Journal of Mathematics* 48, no. 2 (1996): 225-43. doi:10.4153/CJM-1996-011-5.