Some Aspects of Function Theory for Dirichlet-type Spaces

Author

Shuaibing Luo, University of Tennessee - KnoxvilleFollow

Date of Award

8-2014

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Stefan Richter

Committee Members

Carl Sundberg, Remus Nicoara, George Siopsis

Abstract

Let μ[mu] be a nonnegative Borel measure on the boundary T[unit circle] of the unit disc and define φμ[phi mu] to be the harmonic function

φμ(z)= ∫[integral]_T (1-|z|²[square])/(|ζ[zeta]-z|²) dμ(ζ ).

The harmonically weighted Dirichlet space D(μ) is defined as the space of all analytic functions on the unit disc D[unit disc] such that

∫[integral]_D |f'(z)|²φμ(z)dA(z)

is finite. When μ is the Lebesgue measure on T, then D(μ) is the Dirichlet space D.

The harmonically weighted Dirichlet spaces were introduced by Richter in [50] as he was studying analytic two-isometries. These spaces have been studied extensively throughout the years, see e.g. [3], [21], [22], [23], [24], [52], [53], [62], [63], [64], [66] and [67].

The weak product of D denoted by D⊙D [DdotD] is the following set:

D⊙D = {h ∈[in] Hol(D) : h = ∑[sum] figi,

∑[sum] ||fi|| ||gi|| < ∞ [infinity], fi, gi∈D}.

The dual of D⊙D has been characterized in 2010 by Arcozzi, Rochberg, Sawyer and Wick [9] as the space X[script X](D) of analytic functions b on D such that |b'|²dA is a Carleson measure for the Dirichlet space.

In this dissertation we show that for functions f in proper weak*-closed M_z[M sub z]*-invariant subspaces of X(D), the functions (zf)' are in the Nevanlinna class of D and have meromorphic pseudocontinuations in the Nevanlinna class of the exterior disc. We then use this result to show that every nonzero M_z-invariant subspace N[script N] of D⊙D has index 1, i.e. satisfies dim N/zN =1.

In the second part of this dissertation, we study the corona theorem for the D(μ) spaces when μ is a finitely atomic measure. If μ is a finitely atomic measure, we use the observation from Richter and Sundberg [52] that M(D(μ))= D(μ)∩[intersection]H∞(D) to show that the set of multiplicative linear functionals consisting of evaluations at points of D is dense in the maximal ideal space of M(D(μ)). Furthermore, we obtain the corona theorem for infinitely many functions in M(D(μ).

Recommended Citation

Luo, Shuaibing, "Some Aspects of Function Theory for Dirichlet-type Spaces. " PhD diss., University of Tennessee, 2014.
https://trace.tennessee.edu/utk_graddiss/2841