Simply interpolating sequences in complete Pick spaces

TL;DR

This paper characterizes simply interpolating sequences in complete Pick spaces, showing they are exactly the strongly separated sequences, and explores the distinction between simple and multiplier interpolation in various spaces.

Contribution

It provides a complete characterization of simply interpolating sequences in complete Pick spaces and demonstrates the difference between simple and multiplier interpolation in key examples.

Findings

Simply interpolating sequences are exactly the strongly separated sequences.

In many complete Pick spaces, simple interpolation does not imply multiplier interpolation.

Constructs examples of simply interpolating sequences that generate infinite measures.

Abstract

We characterize simply interpolating sequences (also known as onto interpolating sequences) for complete Pick spaces. We show that a sequence is simply interpolating if and only if it is strongly separated. This answers a question of Agler and McCarthy. Moreover, we show that in many important examples of complete Pick spaces, including weighted Dirichlet spaces on the unit disc and the Drury-Arveson space in finitely many variables, simple interpolation does not imply multiplier interpolation. In fact, in those spaces, we construct simply interpolating sequences that generate infinite measures, and uniformly separated sequences that are not multiplier interpolating.