Singular values of weighted composition operators and second quantization

TL;DR

This paper investigates the singular values of weighted composition operators on Hardy spaces, providing bounds and estimates motivated by conformal field theory, and explores their relation to restriction operators and a Fisher-Micchelli phenomenon.

Contribution

It establishes bounds on singular values of weighted composition operators and links these to restriction operators, introducing an analog of the Fisher-Micchelli phenomenon for non-compact operators.

Findings

Bounds on singular values of weighted composition operators.

Estimates on singular values of restriction operators.

Identification of an analog of the Fisher-Micchelli phenomenon.

Abstract

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.