Continuity conditions for the Hilbert transform on quasi-Hilbert spaces (Q2875793)

Let \(X\) be a complex quasi-Hilbert space, i.e., a complex, reflexive, strictly convex Banach space with a Gâteaux-differentiable norm and a quasi-inner product. For a strongly continuous group of operators \((U(t))_{t\in\mathbb{R}}\subset B(X)\), the Hilbert transform \(H\) is defined by NEWLINE\[NEWLINEHx:=\lim_{\varepsilon\to 0, N\to\infty} \frac{1}{\pi}\int_{\varepsilon\leq |t|\leq N} \frac{U(t)x}{x} dt,\quad x\in X,NEWLINE\]NEWLINE see also [\textit{S. Ishikawa}, Tokyo J. Math. J. 9, 383--393 (1986; Zbl 0629.47034)].NEWLINENEWLINEThe paper essentially consists of the proof of the following result: Let \((U(t))_{t\in\mathbb{R}}\) be a group of isometries with the generator \(iA\) and let \(A_+\) be the positive square root of \(-A^2\). Then, \(H\in B(X)\) iff \(iA_+\) generates a bounded strongly continuous operator group in \(B(X)\).NEWLINENEWLINEThe proof strongly relies on results from the first two authors [Novi Sad J. Math. 35, No. 2, 41--55 (2005; Zbl 1263.47054)].