Maximal contractive tuples

TL;DR

This paper investigates the concept of maximality in contractive operator tuples and submodules of the Drury-Arveson module, providing characterizations and contrasting properties between one-dimensional and higher-dimensional cases.

Contribution

It introduces a new notion of maximality for submodules of the Drury-Arveson module and characterizes when such submodules are maximal, highlighting differences across dimensions.

Findings

Every submodule of the Hardy module over the unit disc is maximal.

Homogeneous submodules or those generated by polynomials are not maximal for dimensions d ≥ 2.

A characterization criterion for maximal submodules is established.

Abstract

Maximality of a contractive tuple of operators is considered. Characterization of a contractive tuple to be maximal is obtained. Notion of maximality of a submodule of Drury-Arveson module on the $d$-dimensional unit ball $\mathbb{B}_d$ is defined. For $d=1$, it is shown that every submodule of the Hardy module over the unit disc is maximal. But for $d\ge 2$ we prove that any homogeneous submodule or submodule generated by polynomials is not maximal. A characterization of a submodule to be maximal is obtained.