Hilbert transforms on one parameter groups of operators (Q1093877)

The author generalizes \textit{M. Cotlar}'s Hilbert transform [Rev. Mat. Cuyana 1, 105-167 (1956; Zbl 0071.334)] to one-parameter groups \((U_ t)_{t\in {\mathbb{R}}}\) of linear operators on a complete locally convex vector space X by defining \[ Hx:=\lim_{\epsilon \to 0+, n\to \infty}(1/\pi)\int_{\epsilon <| t| <n}t^{-1}U_ tx dt \] for \(x\in X\) (whenever the limit exists). He shows that under certain conditions (e.g., X Hilbert, \(U_ t\) unitary) H is continuous on X. Part two starts with the Hilbert space situation. It is shown that H satisfies \(H(Hx)=-(x-\bar x)\) for x,y\(\in X\) and \[ \bar x:=\lim_{t\to \infty}(1/2t)\int^{t}_{-t}U_ tx dt. \] For locally convex spaces this remains true as long as \(x\in D(H^ 2)\).