I am trying to solve the Dirichlet problem on the disk: find a square integrable, holomorphic function f on the unit disk such that Re f has a prescribed boundary value x in $L^2(\mathbb{T})$, when restricted to the circle. Furthermore, show that f is unique up to an additive, imaginary constant.
So far I have considered the orthonormal basis $\{e_n : n \in \mathbb{Z}\}$, where each $e_n$ is defined by \begin{align*} e_n := (2\pi)^{-1/2}z^n \ \forall \ z \in \mathbb{T}, \end{align*} on $L^2(\mathbb{T})$. I have shown that the operator $W \in {\bf B}(L^2(\mathbb{T}))$ defined by \begin{align*} We_0 = e_0, \ \ \ We_n = -ie_n, \ \ \ We_{-n} = ie_{-n} \ \forall \ \mathbb{N} \end{align*} is a unitary operator that maps real functions to real functions in $L^2(\mathbb{T})$.
I am not sure how to start solving the Dirichlet problem, i.e., I am not sure how to look for such a function f and how to use the above observations in this search.
Let $b \in L^2(\mathbb{T})$ be the prescribed real part of the boundary value.
Let $$a_n=\frac{1}{2\pi}\int_0^{2\pi}{b(e^{it})e^{-int}\,dt}$$ be its Fourier coefficients, $a_n \in \ell^2$ and $a_{-n}=\overline{a_n}$.
Consider a real function $h \in L^2(\mathbb{T})$, of which the Fourier coefficients are $ia_n$ for $n < 0$ and $-ia_n$ for $n > 0$, then $b+ih \in L^2(\mathbb{T})$ and has no nonzero Fourier coefficients with negative indices. Denote $c_n$ for $n \geq 0$ its Fourier coefficients (with the same formula as above), $c_n \in \ell^2$.
Define then $f(z)=\sum_n{c_nz^n}$ on the unit disk. You can check that $f$ is $L^2$ and that its radial limit (in $L^2(\mathbb{T})$ and a.e.) is $b+ih$.