Let $a,b>0$, $F$ be Fourier transformation and $\chi$ be indicator function. Suppose $T:L^2(\mathbb{R}) \to L^2(\mathbb{R})$ $Tf(y):=F^{-1}(F(f\chi _{[-a,a]})\chi _{[-b,b]})$ and $\{c_n\}_{n=1}^{\infty}$ are all eigenvalues except $0$. Then prove $\sum_{n=1}^{\infty} |c_n|^2 \leq 4ab$
My idea : I proved $T$ is compact. Let $P_n$ be a projection to eigenspace of $c_n$. By spectral theorem, $Tf=\sum _{n=1}^{\infty}c_n P_nf$, so $||Tf||^2=\sum |c_n|^2||P_nf||^2$. But I can't prove $\sum_{n=1}^{\infty} |c_n|^2 \leq 4ab$