Spectra of the translations and Wiener-Hopf operators on $L_\omega ^2({\mathbb R}^+)$

Abstract:

We study bounded operators $T$ on the weighted space $L^2_{\omega }(\mathbb {R}^+)$ commuting either with the “right shift operators” $(R _t)_{t \geq 0}$ or “left shift operators” $(L_{-t})_{t \geq 0},$ and we establish the existence of a symbol $\mu$ of $T$. We characterize completely the spectrum $\sigma (R_t)$ of the operator $R_t$ proving that \[ \sigma (R _t) = \{z \in \mathbb {C}: |z| \leq e^{\alpha _0 t}\},\] where $\alpha _0$ is the growth bound of $(R_t)_{t\geq 0}$. We obtain a similar result for the spectrum of $L_{-t},\: t >0.$ Moreover, for a bounded operator $T$ commuting with $R _t, \: t \geq 0,$ we establish the inclusion $\overline {\mu ({\mathcal O})}\subset \sigma (T)$, where \[ \mathcal {O}= \{ z \in \mathbb {C}: \operatorname {Im} z < \alpha _0\}.\]