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 Lpa(G \ K), 1 ≤ p < ∞, which contain Lpa(G) and with dim(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 Lpa(G) and Lpa (G \ K). When p = ∞, we will remark on the structure of the weak-star closed z-invariant subspaces between H(G) and H(G \ K). When G \ K is not connected, we will show that in general the invariant subspaces between Lpa (G) and Lpa(G \ K) are fantastically complicated. As an application of these results, we will remark on the complexity of the invariant subspaces (under f-> Cf) of certain Besov spaces on K. In particular, we shall see that in the harmonic Dirichlet space B12(T), there are invariant subspaces F such that the dimension of (F in F is infinite.

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

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