The Finite Hilbert Transform in ℒ2

It is well known that the finite HILBERT transform T is a NOETHER (FREDHOLM) operator when considered as a map from ?^p into itself if 1 < p < 2 or 2 < p < ∞. When p = 2, the map T is not a NOETHER operator. We present two theorems which characterize the range of T in ?² and, as immediate consequences, give simple expressions for its inverse.