Pluricapacity and approximation numbers of composition operators

TL;DR

This paper provides estimates for the approximation numbers of composition operators on certain complex domains, linking these estimates to the Monge-Ampère capacity of the image of the domain under the symbol function.

Contribution

It introduces new bounds for approximation numbers of composition operators on hyperconvex sets, connecting operator theory with pluripotential theory.

Findings

Approximation numbers are estimated using Monge-Ampère capacity.

Results apply to hyperconvex sets like the ball and polydisk.

Provides bounds for composition operators with relatively compact images.

Abstract

For suitable bounded hyperconvex sets $\Omega$ in $\mathbb{C}^N$, in particular the ball or the polydisk, we give estimates for the approximation numbers of composition operators $C_\phi \colon H^2 (\Omega) \to H^2 (\Omega)$ when $\phi (\Omega)$ is relatively compact in $\Omega$, involving the Monge-Amp\`ere capacity of $\phi (\Omega)$.