Invariant Subspaces in Spaces of Analytic Functions

Abstract: Let D be a finitely connected bounded domain with smooth boundary in the complex plane. We first study Banach spaces of analytic functions on D . The main result is a theorem which converts the study of hyperinvariant subspaces on multiply connected domains into the study of hyperinvariant subspaces on domains with fewer holes. The Banach spaces are defined by a natural set of axioms fulfilled by the familiar Hardy, Dirichlet, and Bergman spaces. Let D_1 be a bounded domain obtained from D by adding some of the connectivity components of the complement of D ; hence D_1 has fewer holes. Let B and B_1 be the Banach spaces of analytic functions on the domains D and D_1 , respectively. Assume that I is a hyperinvariant subspace of B_1 , and consider the smallest hyperinvariant subspace of B containing I ; this is Lambda (I) , the closure in B of the span of I cdot M(B) , where M(B) denotes the space of multipliers of B . Under reasonable assumptions, we prove that I mapsto Lambda (I) gives a one-to-one correspondence between a class of hyperinvariant subspaces of B_1 , and a class of hyperinvariant subspaces of B . The inverse mapping is given by J mapsto J cap B_1 . We then generalize the above result to the setting of the quasi-Banach spaces of analytic functions on D . In Chapter IV, we shall establish a Riesz-type representation formula for super-biharmonic functions satisfying certain growth conditions on the unit disk. This representation formula can be regarded as an analogue of the Poisson-Jensen representation formula for subharmonic functions. In Chapter V, this representation formula will be used to prove an approximation theorem in certain weighted Bergman spaces. More precisely, we consider those weighted Bergman spaces whose (non-radial) weights are super-biharmonic and fulfill a certain growth condition. We shall prove that the polynomials are dense in such weighted Bergman spaces.