## 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 π_{Ξ±} [script D sub alpha] are the family of Hilbert spaces consisting of all analytic functions f(z) = β_{n=0}^{β}[sum over n equals zero to infinity] Λf(n) [f hat of n] z^{n} [z to the n] defined on the open unit disc π» [unit disc] such that

β_{n=0}^{β} (n+ 1)^{Ξ±} |Λf(n )|^{2}

is finite.

For Ξ± < 0, the spaces π_{Ξ±} are known as weighted Bergman spaces. When Ξ±= 0, then π_{0}= H^{2}, the well known and much studied Hardy space. For Ξ± > 0, the π_{Ξ±} spaces are weighted Dirichlet spaces.

The characterization of the invariant subspaces of the multiplication operator M_{z} [M sub z] on the π_{Ξ±} spaces depends on Ξ±, and it is partially still an open problem. The invariant subspaces of π_{2} have been characterized in 1972 by B. I. Korenblum [25].

In this dissertation we show that the invariant subspaces of π_{2} can be approximated by finite co-dimensional invariant subspaces. For the Dirichlet space D= π_{1} there is no complete characterization of invariant subspaces, but we consider

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

where E β [subset] π [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 π the square of the absolute value of Ο converges to its boundary value in certain tangential approach regions.

#### Recommended Citation

Yilmaz, Faruk, "Approximation of Invariant Subspaces. " PhD diss., University of Tennessee, 2017.

https://trace.tennessee.edu/utk_graddiss/4672