Bergman Spaces on Disconnected Domains

Aleman, Alexandru ; Richter, Stefan ; Ross, William T.

Canadian journal of mathematics, Tome 48 (1996) no. 2, pp. 225-243

Résumé

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 → zƒ) of the Bergman space (G \ K), 1 ≤ p < ∞, which contain (G) and with dim(M/(z - λ)M) = 1 for all λ ∈ G \ K. When G \ K is connected, we will see that dim(M/(z - λ)M) = 1 for all λ ∈ G \ K and thus in this case we will have a complete description of the invariant subspaces lying between (G) and (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 (G) and (G \ K) are fantastically complicated. As an application of these results, we will remark on the complexity of the invariant subspaces (under ƒ → ζƒ) of certain Besov spaces on K. In particular, we shall see that in the harmonic Dirichlet space , there are invariant subspaces F such that the dimension of ζF in F is infinite.

Détail

DOI : 10.4153/CJM-1996-011-5

Mots-clés : 47B38, 46E15, 46E35, Bergman spaces, invariant subspaces, duality, Sobolev spaces, Besov spaces, Hausdorff measure, capacity

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