Summary: The truncated Fourier operator \(\mathcal {F}_{\mathbb{R}^+}\), \[(\mathcal {F}_{\mathbb{R}^+}x)(t)=\frac {1}{\sqrt {2\pi }}\int _{\mathbb{R}^+}x(\xi )e^{it\xi }\,d\xi ,\quad t\in {\mathbb{R}^+},\] is studied. The operator \(\mathcal {F}_{\mathbb{R}^+}\) is viewed as an operator acting in the space \(L^2(\mathbb{R}^+)\). A functional model for the operator \(\mathcal {F}_{\mathbb{R}^+}\) is constructed. This functional model is the operator of multiplication by an appropriate (\({2\times 2}\))-matrix function acting in the space \(L^2(\mathbb{R}^+)\oplus L^2(\mathbb{R}^+)\). Using this functional model, the spectrum of the operator \(\mathcal {F}_{\mathbb{R}^+}\) is found. The resolvent of the operator \(\mathcal {F}_{\mathbb{R}^+}\) is estimated near its spectrum.