A functional model for the Fourier--Plancherel operator truncated on the positive half-axis

Abstract

The truncated Fourier operator $\mathscr{F}_{\mathbb{R^{+}}}$, $$ (\mathscr{F}_{\mathbb{R^{+}}}x)(t)=\frac{1}{\sqrt{2\pi}} \int\limits_{\mathbb{R^{+}}}x(\xi)e^{it\xi}\,d\xi\,,\ \ \ t\in{}{\mathbb{R^{+}}}, $$ is studied. The operator $\mathscr{F}_{\mathbb{R^{+}}}$ is considered as an operator acting in the space $L^2(\mathbb{R^{+}})$. The functional model for the operator $\mathscr{F}_{\mathbb{R^{+}}}$ is constructed. This functional model is the multiplication operator on the 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.