Consider the following operator on functions $\mathcal{T}: L^2(0,1) \to L^2(0,1)$ over the complex numbers. \begin{equation} (\mathcal{T} f)(z_1) = \mathrm{p.v.} \int_0^1 \frac{i}{z_1-z_2}f(z_2) dz_2 \end{equation} where $\mathrm{p.v.}$ means the Cauchy principal value. This is an analogue of the Hilbert Transform except restricted to functions on an interval.

Note that this operator is Hermitian, so there should be an orthogonal basis of eigenfunctions. Is there a known description of such an orthogonal basis of the eigenfunctions and eigenvalues of this operator and a relevant 'Fourier inversion' formula?