Linear invariance and extension operators of Pfaltzgraff-Suffridge type

"We consider the image of a linear-invariant family $\mathcal{F}$ of normalized locally biholomorphic mappings defined in the Euclidean unit ball $\B_n$ of $\C^n$ under the extension operator $$\Phi_{n,m,\beta}[f](z,w) = \mleft( f(z), [Jf(z)]^\beta w\mright), \quad (z,w) \in \B_{n+m} \subseteq \C^n \times \C^m,$$ where $\beta \in \C$, $Jf$ denotes the Jacobian determinant of $f$, and the branch of the power function taking $0$ to $1$ is used. When $\beta=1/(n+1)$ and $m=1$, this is the Pfaltzgraff--Suffridge extension operator. In particular, we determine the order of the linear-invariant family on $\B_{n+m}$ generated by the image in terms of the order of $\mathcal{F}$, taking note that the resulting family has minimum order if and only if either $\beta \in (-1/m,1/(n+1)]$ and the family $\mathcal{F}$ has minimum order or $\beta=-1/m$. We will also see that order is preserved when generating a linear-invariant family from the family obtained by composing $\mathcal{F}$ with a certain type of automorphism of $\C^n$, leading to consequences for various extension operators including the modified Roper--Suffridge extension operator introduced by the author."