Doctoral Dissertations

Date of Award

8-2017

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Stefan Richter

Committee Members

Carl Sundberg, Michael Frazier, Michael W. Berry

Abstract

For a real number Ξ± [alpha] the Dirichlet-type spaces 𝔇α [script D sub alpha] are the family of Hilbert spaces consisting of all analytic functions f(z) = βˆ‘n=0∞[sum over n equals zero to infinity] Λ†f(n) [f hat of n] zn [z to the n] defined on the open unit disc 𝔻 [unit disc] such that

βˆ‘n=0∞ (n+ 1)Ξ± |Λ†f(n )|2

is finite.

For Ξ± < 0, the spaces 𝔇α are known as weighted Bergman spaces. When Ξ±= 0, then 𝔇0= H2, the well known and much studied Hardy space. For Ξ± > 0, the 𝔇α spaces are weighted Dirichlet spaces.

The characterization of the invariant subspaces of the multiplication operator Mz [M sub z] on the 𝔇α spaces depends on Ξ±, and it is partially still an open problem. The invariant subspaces of 𝔇2 have been characterized in 1972 by B. I. Korenblum [25].

In this dissertation we show that the invariant subspaces of 𝔇2 can be approximated by finite co-dimensional invariant subspaces. For the Dirichlet space D= 𝔇1 there is no complete characterization of invariant subspaces, but we consider

DE= {f ∈ [in]D : f* = 0 q.e. [quasi-everywhere] on E}

where E βŠ† [subset] 𝕋 [unit circle] is a Carleson thin set. In this case, we have a partial result.

In the second part of the dissertation we prove a regularity result for extremal functions in the Dirichlet space D. If Ο† [phi] is an extremal function in the Dirichlet space, then we use a result of Richter and Sundberg [35] to show that for each point on the unit circle 𝕋 the square of the absolute value of Ο† converges to its boundary value in certain tangential approach regions.

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