  We study a semigroup of weighted composition operators on the Hardy space of
the disk $H^2(\mathbb{D})$, and more generally on the Hardy space $H^2(U)$
attached to a simply connected domain $U$ with smooth boundary. Motivated by
conformal field theory, we establish bounds on the singular values
(approximation numbers) of these weighted composition operators. As a byproduct
we obtain estimates on the singular values of the restriction operator
(embedding operator) $H^2(V) \to H^2(U)$ when $U \subset V$ and the boundary of
$U$ touches that of $V$. Moreover, using the connection between the weighted
composition operators and restriction operators, we show that these operators
exhibit an analog of the Fisher-Micchelli phenomenon for non-compact operators.
