 Solution

Approximate $h$ uniformly by finite sums $\sum f_j(x)g_j(y)$; their integral operators have finite-dimensional range and converge in operator norm, so $T_h$ is compact. If $h(x,y)=0$ for $x\le y$, iterated integration over an ordered simplex gives $\Vert T_h^n\Vert \le \Vert h{\Vert }_{\infty }^n/n!$. Thus $r(T_h)=0$, and compact spectral theory gives $\sigma (T_h)=\{0\}$.