On the chain structure in the de Branges spaces

TL;DR

This paper investigates the structure of chains of de Branges subspaces, focusing on indivisible intervals and growth properties related to spectral measures supported on integers, revealing limitations on subspace types.

Contribution

It establishes that for spectral measures on integers, there are at most two subspaces of the same type bounding an indivisible interval, and analyzes their possible locations.

Findings

At most two subspaces of the same type bound an indivisible interval for spectral measures on integers.

Characterization of the locations of indivisible intervals in these chains.

Insights into the growth of exponential type in de Branges subspace chains.

Abstract

We study the indivisible intervals and the monotonicity of the growth of the exponential type in the chains of de Branges subspaces in terms of the spectral measure. We prove that for spectral measures supported on $\mathbb Z$, there exist at most two subspaces of the same type, which then bound an indivisible interval. Furthermore, in this case, we study possible locations of the indivisible intervals.