A de Branges-Beurling theorem for the full Fock space

TL;DR

This paper generalizes the de Branges-Beurling theorem to the full Fock space, a non-commutative Hardy space, and explores lattice operations and multipliers within this framework, advancing operator theory in non-commutative analysis.

Contribution

It extends classical shift-invariant space characterizations to non-commutative Fock spaces and analyzes lattice structures of operator-valued multipliers.

Findings

Characterization of shift-invariant spaces in non-commutative Fock space

Lattice structure of operator-valued multipliers

Boundedness of multipliers with common coefficient range

Abstract

We extend the de Branges-Beurling theorem characterizing the shift-invariant spaces boundedly contained in the Hardy space of square-summable power series to the full Fock space over $\mathbb{C} ^d$. Here, the full Fock space is identified as the \emph{Non-commutative (NC) Hardy Space} of square-summable Taylor series in several non-commuting variables. We then proceed to study lattice operations on NC kernels and operator-valued multipliers between vector-valued Fock spaces. In particular, we demonstrate that the operator-valued Fock space multipliers with common coefficient range space form a bounded general lattice modulo a natural equivalence relation.