The truncated Fourier operator. General results

Let $\mathcal F$ be the one dimensional Fourier–Plancherel operator and $E$ be a subset of the real axis. The truncated Fourier operator is the operator $\mathcal F_E$ of the form $\mathcal F_E=P_E\mathcal FP_E$, where $(P_Ex)(t)=\mathbf 1_E(t)x(t)$, and $\mathbf 1_E(t)$ is the indicator function of the set $E$. In the presented work, the basic properties of the operator $\mathcal F_E$ according to the set $E$ are discussed. Among these properties there are the following ones:
1) the operator $\mathcal F_E$ has a nontrivial null-space;
2) $\mathcal F_E$ is strictly contractive;
3) $\mathcal F_E$ is a normal operator;
4) $\mathcal F_E$ is a Hilbert–Schmidt operator;
5) $\mathcal F_E$ is a trace class operator.