  We study the approximation numbers of weighted composition operators
$f\mapsto w\cdot(f\circ\varphi)$ on the Hardy space $H^2$ on the unit disc. For
general classes of such operators, upper and lower bounds on their
approximation numbers are derived. For the special class of weighted lens map
composition operators with specific weights, we show how much the weight $w$
can improve the decay rate of the approximation numbers, and give sharp upper
and lower bounds. These examples are motivated from applications to the
analysis of relative commutants of special inclusions of von Neumann algebras
appearing in quantum field theory (Borchers triples).
