Approximation of Invariant Subspaces

For a real number α [alpha] the Dirichlet-type spaces 𝔇_α [script D sub alpha] are the family of Hilbert spaces consisting of all analytic functions f(z) = ∑_n=0^∞[sum over n equals zero to infinity] ˆf(n) [f hat of n] zⁿ [z to the n] defined on the open unit disc 𝔻 [unit disc] such that

∑_n=0^∞ (n+1)^α | ˆf(n) |²

[sum over n equals 0 to infinity] [(n+1) to α] [ | f hat of n | to 2]

is finite.

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

The characterization of the invariant subspaces of the multiplication operator M_z [M sub z] on the 𝔇_α spaces depends on α, 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

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

[D subscript E] [equals] [{f[in]D: [f superscript *] [equals 0] [quasi-everywhere] [on E]}]

where E ⊆ [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 φ [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 φ converges to its boundary value in certain tangential approach regions.