# On a class of shift-invariant subspaces of the Drury-Arveson space

## TL;DR

This paper studies a specific class of shift-invariant subspaces in the Drury-Arveson space, providing explicit kernel formulas, characterizations via Hankel operators, and applications to operator inequalities.

## Contribution

It introduces a new class of shift-invariant subspaces with explicit kernels and Hankel operator characterizations, and adapts Drury's inequality to these spaces.

## Key findings

- Explicit kernel formulas for the subspaces.
- Characterization as intersections of Hankel operator kernels.
- Optimal adaptation of Drury's inequality.

## Abstract

In the Drury-Arveson space, we consider the subspace of functions whose Taylor coefficients are supported in the complement of a set $Y\subset\mathbb{N}^d$ with the property that $Y+e_j\subset Y$ for all $j=1,\dots,d$. This is an easy example of shift-invariant subspace, which can be considered as a RKHS in is own right, with a kernel that can be explicitely calculated. Moreover, every such a space can be seen as an intersection of kernels of Hankel operators, whose symbols can be explicity calcuated as well. Finally, this is the right space on which Drury's inequality can be optimally adapted to a sub-family of the commuting and contractive operators originally considered by Drury.