Estimates for approximation numbers of some classes of composition operators on the Hardy space

TL;DR

This paper provides estimates for the approximation numbers of composition operators on the Hardy space, relating them to the properties of the symbol function, with applications to specific maps like the cusp map.

Contribution

It introduces new bounds for approximation numbers based on the symbol's boundary behavior and extends existing results to operators with symbols touching the boundary at finitely many points.

Findings

Approximation numbers decay exponentially for symbols mapping into polygons.

Improved upper bounds for symbols with boundary contact points.

Existence of compact composition operators with prescribed boundary behavior.

Abstract

We give estimates for the approximation numbers of composition operators on $H^2$, in terms of some modulus of continuity. For symbols whose image is contained in a polygon, we get that these approximation numbers are dominated by $\e^{- c \sqrt n}$. When the symbol is continuous on the closed unit disk and has a domain touching the boundary non-tangentially at a finite number of points, with a good behavior at the boundary around those points, we can improve this upper estimate. A lower estimate is given when this symbol has a good radial behavior at some point. As an application we get that, for the cusp map, the approximation numbers are equivalent, up to constants, to $\e^{- c \, n / \log n}$, very near to the minimal value $\e^{- c \, n}$. We also see the limitations of our methods. To finish, we improve a result of O. El-Fallah, K. Kellay, M. Shabankhah and H. Youssfi, in showing that for every compact set $K$ of the unit circle $\T$ with Lebesgue measure 0, there exists a compact composition operator $C_\phi \colon H^2 \to H^2$, which is in all Schatten classes, and such that $\phi = 1$ on $K$ and $|\phi | < 1$ outside $K$.