Cyclic vectors in spaces of analytic functions
The thesis consists of two parts. Part One of the thesis is devoted to the study of the Dirichlet space Dir(G) for finitely connected regions G; the author is particularly interested in the algebra of multiplication operators on this space. Results in different directions are obtained. One direction deals with the structure of closed subspaces invariant under all multiplication operators. In particular, he shows that each such a subspace contains a bounded function. For regions with circular boundaries, he proves that a finite codimensional closed subspace invariant under multiplication by z must be invariant under all multiplication operators, and furthermore it is of the form pDir(G), where p is a polynomial with all its roots lying in G. Another direction is to study cyclic and noncyclic vectors for the algebra of all multiplication operators. Typical results are: if f is a function in Dir(G) and it is bounded away from zero then f is cyclic; on the other hand, if the zero set of the radial limit function of f on the boundary has positive logarithmic capacity, then f is not cyclic. Also, some other sufficient conditions for a function to be cyclic are given. Lastly, he studies transitive operator algebras containing all multiplication operators; he proves that they are dense in the algebra of all bounded operators in the strong operator topology. Part Two of the thesis studies common cyclic vectors. The existence of vectors that are cyclic for the adjoint of any nonscalar multiplication operator on the Hardy space H{sup 2} was proved by Wogen. Furthermore, he showed that all these common cyclic vectors form a dense set in H{sup 2}. The author extends these results to quite general Banach spaces of analytic functions in bounded domains in the complex plane.
- Research Organization:
- Michigan Univ., Ann Arbor, MI (USA)
- OSTI ID:
- 5903122
- Resource Relation:
- Other Information: Thesis (Ph. D.)
- Country of Publication:
- United States
- Language:
- English
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