A remark on the spectrum of the analytic generator. (Q5960842)

For a one-parameter group \(U=(U_t)_{t\in\mathbb R}\) of isometries on a Banach space (with a number of additional properties), the analytic generator \(U_i\) is defined as the analytic extension of \(U\) (for suitable elements) to the point \(i\in \mathbb C\). In general, the spectrum of \(U_i\) may by all of \(\mathbb C\), but it is shown that the `amplification' of \(U_i\) to its graph has its spectrum in \(\mathbb R_+\).