# Row contractions annihilated by interpolating vanishing ideals

## TL;DR

This paper explores the structure of commuting row contractions annihilated by higher order vanishing ideals, providing a Jordan-type decomposition, characterizing interpolating sequences, and classifying certain contractions up to quasi-similarity.

## Contribution

It introduces a Jordan-type decomposition for these contractions, characterizes interpolating sequences operator-theoretically, and refines classification results up to quasi-similarity.

## Key findings

- Jordan-type direct sum decomposition for row contractions
- Operator-theoretic characterization of interpolating sequences
- Classification of cyclic contractions up to quasi-similarity

## Abstract

We study similarity classes of commuting row contractions annihilated by what we call higher order vanishing ideals of interpolating sequences. Our main result exhibits a Jordan-type direct sum decomposition for these row contractions. We illustrate how the family of ideals to which our theorem applies is very rich, especially in several variables. We also give two applications of the main result. First, we obtain a purely operator theoretic characterization of interpolating sequences. Second, we classify certain classes of cyclic commuting row contractions up to quasi-similarity in terms of their annihilating ideals. This refines some of our recent work on the topic. We show how this classification is sharp: in general quasi-similarity cannot be improved to similarity. The obstruction to doing so is the existence, or lack thereof, of norm-controlled similarities between commuting tuples of nilpotent matrices, and we investigate this question in detail.