A functional model for the Fourier–Plancherel operator truncated to the positive semiaxis

The truncated Fourier operator $\mathscr F_{\mathbb R^+}$,
\begin{equation*} (\mathscr F_{\mathbb R^+}x)(t)=\frac1{\sqrt{2\pi}}\int_{\mathbb R^+}x(\xi)e^{it\xi}\,d\xi,\quad t\in\mathbb{R^+}, \end{equation*}
is studied. The operator $\mathscr F_{\mathbb R^+}$ is viewed as an operator acting in the space $L^2(\mathbb R^+)$. A functional model for the operator $\mathscr F_{\mathbb R^+}$ is constructed. This functional model is the operator of multiplication by an appropriate ($2\times2$)-matrix function acting in the space $L^2(\mathbb R^+)\oplus L^2(\mathbb R^+)$. Using this functional model, the spectrum of the operator $\mathscr F_{\mathbb R^+}$ is found. The resolvent of the operator $\mathscr F_{\mathbb R^+}$ is estimated near its spectrum.