Approximation numbers of weighted composition operators

TL;DR

This paper investigates the approximation numbers of weighted composition operators on the Hardy space, providing bounds and analyzing how weights influence decay rates, with applications in quantum field theory.

Contribution

It offers new bounds on approximation numbers for weighted composition operators and explores how weights affect decay rates, with specific focus on weighted lens map operators.

Findings

Derived upper and lower bounds for approximation numbers.

Showed how weights can improve decay rates of approximation numbers.

Connected results to applications in quantum field theory.

Abstract

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).