Let $\Lambda \subset \mathbb{R}$ be a bounded domain, and let $T:L^2(\Lambda)\to L^2(\Lambda)$ be a compact operator. Can we say anything about the regularity of $Tf$, $f\in L^2(\Lambda)$? Does a compact operator improve the regularity of the function $f$ in this case? What about the case where $T$ is trace-class or Hilbert-Schmidt?
Thank you very much! Luke
If $f_0$ is a fixed function in $L^{2}$ then $Tf=\langle f , f_0 \rangle f_0$ is compact, of trace class and Hilbert-Shmidt. $Tf$ does not have any smoothness property if $f_0$ does not have such a property except when $Tf=0$ (or $\langle f , f_0 \rangle=0$).