Surjectivity of isometries of weighted spaces of holomorphic functions and of Bloch spaces

TL;DR

This paper investigates the conditions under which isometries of weighted holomorphic function spaces and Bloch spaces are surjective, providing criteria and showing surjectivity for classical weights and specific spaces.

Contribution

It establishes surjectivity of isometries in weighted holomorphic spaces and Bloch spaces, with new criteria based on separation conditions and classical weights.

Findings

All isometries of certain weighted spaces on the unit disc are surjective.

Criteria for surjectivity of isometries in weighted spaces are provided.

All isometries of the little Bloch space are surjective.

Abstract

We examine the surjectivity of isometries between weighted spaces of holomorphic functions. We show that for certain classical weights on the open unit disc all isometries of the weighted space of holomorphic functions, ${ \mathcal H}_{v_o}( \Delta)$, are surjective. Criteria for surjectivity of isometries of ${ \mathcal H}_v(U)$ in terms of a separation condition on points in the image of ${ \mathcal H}_{v_o}(U)$ are also given for $U$ a bounded open set in $\mathbb{C}$. Considering the weight $v(z)= 1-|z|^2$ and the isomorphism $f\mapsto f'$ we are able to show that all isometries of the little Bloch space are surjective.