The order of a linearly invariant familiy in C^n

TL;DR

This paper investigates the trace norm of linearly invariant families in complex n-dimensional space, deriving an upper bound based on the Schwarzian norm and dimension, extending one-variable methods to multiple dimensions.

Contribution

It introduces a new upper bound for the trace norm in C^n by adapting one-variable techniques, linking it to the Schwarzian norm and dimension.

Findings

Derived an upper bound for the trace norm in C^n.

Connected the trace norm to the Schwarzian norm and dimension.

Extended one-variable methods to several complex variables.

Abstract

We study the (trace) norm of a linearly invariant family in the ball in $\C$. By adapting an approach that in one variable yields optimal results, we are able to derive an upper bound for the norm of the family in terms of the Schwarzian norm and the dimension $n$.