Singular Values of Weighted Composition Operators and Second Quantization

We study a semigroup of weighted composition operators on the Hardy space of the disk H²(D), and more generally on the Hardy space H²(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) H²(V) → H²(U) when U ⊂ 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.