Quasiconformal symbols and projected composition operators

TL;DR

This paper investigates the invertibility of projected composition operators with quasiconformal symbols on weighted Bergman spaces, revealing conditions under which invertibility is preserved, especially with small Beltrami coefficients.

Contribution

It introduces new invertibility criteria for projected composition operators with quasiconformal symbols on weighted Bergman spaces, extending known results for conformal symbols.

Findings

Invertibility holds if the Beltrami coefficient is sufficiently small.

Invertibility persists for symbols conformal in an annulus.

Results apply to both standard and exponentially decreasing weights.

Abstract

We study projected composition operators K_g with quasiconformal symbols g on weighted Bergman spaces on the open unit disc D. If the symbol were conformal, i.e.a M\"obius transform of D, the corresponding composition operator would be automatically invertible at least in standard weighted spaces. We show that the invertibility remains, if the Beltrami coefficient is small enough, in particular, it satisfies a certain vanishing condition at the boundary of the disc. We also consider the invertibility of K_g for symbols g which are conformal in an annulus { R < |z| < 1 }. The weight classes in our considerations include both standard and exponentially decreasing weights.