Some Aspects of Function Theory for Dirichlet-type Spaces

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).