Spectrum of $Tf(x)=\int\limits_{-\infty}^\infty \frac{f(y)\,dy}{1+(x-y)^2}$

Question

How can one find spectrum of this $T: L_2(\mathbb{R})\to L_2(\mathbb{R})$?

I kinda hoped that this operator is compact, so that I could look only for point spectrum, but $K(x,y)\notin L_2(\mathbb{R}^2)$, so I'm not sure about it. Also I don't even know how to find point spectrum for this operator.

Answer 1

If $K(x)=\frac{1}{1+x^2}$, then you can write $T$ as the convolution: $$ Tf = K\star f $$ So, the Fourier transform of $Tf$ is $$ \widehat{Tf} = \sqrt{2\pi}\hat{K}\hat{f} $$ Therefore, $T$ is unitarily equivalent to multiplication by $\sqrt{2\pi}\hat{K}$, which is a continuous function vanishing at $\infty$. The spectrum is the range of $\sqrt{2\pi}\hat{K}$ unioned with $\{0\}$.