Composition operators on de Branges spaces of entire functions

TL;DR

This paper investigates the boundedness and compactness of composition operators on de Branges spaces, revealing that boundedness implies affine symbols and highlighting differences from Paley-Wiener spaces.

Contribution

It establishes conditions under which composition operators are bounded or compact on de Branges spaces, including the affine nature of symbols and comparative analysis with Paley-Wiener spaces.

Findings

Boundedness of composition operators implies the inducing symbol is affine.

Affine symbols under certain conditions produce bounded composition operators.

Behavior of these operators differs from that on Paley-Wiener spaces.

Abstract

This paper aims to study the boundedness and compactness of composition operators from model spaces to the Hardy Hilbert spaces in the upper half-plane. Consequently, we investigate the boundedness and compactness of composition operators on de Branges spaces of entire functions. Moreover, we observe that the boundedness of a composition operator on a regular de Branges space forces the inducing symbol to be affine; conversely, affine symbols under appropriate conditions yield bounded composition operators. Furthermore, we show that the behaviour of boundedness and compactness of composition operators on general de Branges spaces is different from that on the Paley-Wiener spaces.