A rigidity theorem for holomorphic generators on the Hilbert ball

TL;DR

This paper proves a rigidity property for holomorphic generators on the Hilbert ball, showing that under certain boundary conditions, the generator must be identically zero.

Contribution

It establishes a new boundary behavior criterion that forces holomorphic generators to vanish identically on the Hilbert ball.

Findings

If the boundary limit of f(z)/||z-τ||^3 is zero, then f is identically zero.

The result applies to generators of one-parameter semigroups on the Hilbert ball.

Provides a boundary rigidity theorem for holomorphic functions in infinite-dimensional settings.

Abstract

We present a rigidity property of holomorphic generators on the open unit ball $\mathbb{B}$ of a Hilbert space $H$. Namely, if $f\in\Hol (\mathbb{B},H)$ is the generator of a one-parameter continuous semigroup ${F_t}_{t\geq 0}$ on $\mathbb{B}$ such that for some boundary point $\tau\in \partial\mathbb{B}$, the admissible limit $K$-$\lim\limits_{z\to\tau}\frac{f(x)}{\|x-\tau\|^{3}}=0$, then $f$ vanishes identically on $\mathbb{B}$.