Integral representation of completely continuous operators in \(L_ 2\) (Q1121504)

The author proved that for every compact operator S in \(L^ 2(0,\infty)\) having an infinite set of characteristic numbers there exists a unitary operator U such that \(USU^*\) and \(US^*U^*\) are integral operators whose kernels K and \(K_*\) verify the Carleman conditions: \[ \int^{\infty}_{0}| K(s,t)|^ 2dt\leq C,\quad for\quad every\quad s\in (0,\infty) \] \[ \lim_{\epsilon \to 0}\int^{\infty}_{0}| K(s+\epsilon,t)-K(s,t)|^ 2dt=0,\quad for\quad every\quad s\in (0,\infty), \] and similarly for \(K_*\).