On approximation numbers of composition operators

TL;DR

This paper investigates the decay rates of approximation numbers of compact composition operators on weighted Bergman spaces, showing they tend to zero at least exponentially and providing bounds and examples.

Contribution

It establishes the minimal exponential decay rate of approximation numbers and offers explicit bounds and examples for these operators.

Findings

Approximation numbers tend to zero at least exponentially.

Decay rate depends on the symbol's approach to the unit circle.

Explicit bounds and examples are provided.

Abstract

We show that the approximation numbers of a compact composition operator on the weighted Bergman spaces $\mathfrak{B}_\alpha$ of the unit disk can tend to 0 arbitrarily slowly, but that they never tend quickly to 0: they grow at least exponentially, and this speed of convergence is only obtained for symbols which do not approach the unit circle. We also give an upper bounds and explicit an example.