Unbounded Bergman operators

Let G be an open subset of the plane, and denote by L²_α(G) the Bergman space of all square integrable analytic functions with respect to the Lebesgue area measure. If dom S_G → {ƒ ∈ L²_α(G): zƒ ∈ L²_α(G), define the Bergman operator S_G: dom S_G → L²_α(G) by S_G: ƒ(z) → zƒ(z). We show that the problem regarding the density of dom SG in L²_α(G) is equivalent to the prob-lem regarding the density of the range of the operator of multiplication by z on some open subset of the unit disc D. If U is a open subset of D containing 0 in its topological boundary, using Wiener capacity we present two suffi-cient conditions such that z[L²_α(U)] is dense in L²_α(U). As a consequence, it follows that if G has finite area, or the component of the complement of G with respect to the extended plane containing ∞ does not equal the single- ton {∞}, then the Bergman operator S_G is densely defined. Furthermore, we prove that if G is a simply connected region of finite area, or a half plane, then the self-commutator of S_G is densely defined. It is also shown that for a Bergman operator S_G, the spectrum equals the closure of G, and its point spectrum is the empty set. Finally, we show that if the Bergman opera-tor S_G is densely defined, then the set of all bounded operators in L²_α(G) that commute with S_G equals {M_φ: φ ∈ H^∞(G)], where M_φ denotes the operator of multiplication by φ on L²_α(G).