Convergence and preservation of cyclicity

TL;DR

This paper investigates how cyclicity properties of functions in Dirichlet-type spaces behave under norm convergence, revealing that cyclicity is not preserved, but some related quantities are.

Contribution

It demonstrates that cyclicity is not preserved under norm limits in Dirichlet-type spaces and analyzes which properties remain stable during convergence.

Findings

Cyclicity is not preserved under norm convergence in $D_\alpha$ spaces.

Certain quantities related to cyclicity are preserved under limits.

Provides a quantitative analysis of convergence effects on cyclicity.

Abstract

We prove the property that a function is cyclic (resp., non-cyclic) is not preserved by norm convergence in Dirichlet-type spaces $D_\alpha$, and show how other significant quantities for cyclicity do remain preserved under the limit of convergent sequences in $D_\alpha$, providing a quantitative view of this convergence issue.