A note on simply interpolating sequences for the Dirichlet space

TL;DR

This paper compares three sufficient conditions for simply interpolating sequences in the Dirichlet space, establishing implications among them and providing counterexamples to reverse implications, thus clarifying their relationships.

Contribution

It analyzes and compares three different sufficient conditions for simply interpolating sequences in the Dirichlet space, clarifying their implications and limitations.

Findings

One box condition implies column bounded property.

Column bounded property implies restricted one box condition.

Counterexamples show reverse implications fail even for weakly separated sequences.

Abstract

We study simply interpolating sequences for the Dirichlet space in the unit disc. In particular we are interested in comparing three different sufficient conditions for simply interpolating sequences. The first one is the the so called one box condition, the second is the column bounded propererty for the associated Grammian matrix and the third one is a restricted version of the one box condition introduced by Bishop and, independently, by Marshall and Sundberg. We prove that the one box condition implies the column bounded property which in turn implies the restricted on box condition of Bishop-Marshall-Sundberg, and we give two counterexamples which show that the reverse implications fail even for weakly separated sequences.