Decay rates for approximation numbers of composition operators

TL;DR

This paper develops a method to estimate the decay rates of approximation numbers for composition operators on Hardy spaces, especially when the symbol map has specific boundary behaviors like smoothness, cusps, or corners.

Contribution

It introduces a general approach using finite-dimensional model subspaces and derives exact or asymptotic decay rates for various boundary regularities of the symbol map.

Findings

Exact decay rate for smooth boundary maps.

Asymptotic decay for cusp boundary maps.

Estimates for corner boundary maps.

Abstract

A general method for estimating the approximation numbers of composition operators on $\Ht$, using finite-dimensional model subspaces, is studied and applied in the case when the symbol of the operator maps the unit disc to a domain whose boundary meets the unit circle at just one point. The exact rate of decay of the approximation numbers is identified when this map is sufficiently smooth at the point of tangency; it follows that a composition operator with any prescribed slow decay of its approximation numbers can be explicitly constructed. Similarly, an asymptotic expression for the approximation numbers is found when the mapping has a sharp cusp at the distinguished boundary point. Precise asymptotic estimates in the intermediate cases, including that of maps with a corner at the distinguished boundary point, are also established.