We investigate the unbounded operator obtained from the finite Hilbert transform when it acts from $L^1(-1, 1)$ into itself. A full inversion theorem is established for this operator, together with suitable extended versions of the Parseval and Poincaré-Bertrand formulae.

The talk is based on joint papers with Oleksiy Karlovych. We consider multidimensional generalisations of Wiener-Hopf operators and show that, under mild restrictions on the underlying Banach function space, their norm is estimated below by the L^infinity norm of the symbol, while their Kuratowski measure of noncompactness is estimated below by half that norm. Additionally, we show that for a classical Wiener-Hopf operator on a (separable) translation-invariant Banach function space over the half-line, its Hausdorff measure of noncompactness, its essential norm, and its norm are all equal.