Contractively embedded invariant subspaces

TL;DR

This paper extends the theory of invariant subspaces to multiple variables, providing a de Branges theorem analogue for contractively embedded subspaces in multivariable Hardy spaces.

Contribution

It introduces a multivariable version of the de Branges theorem for contractively embedded invariant subspaces in analytic reproducing kernel Hilbert spaces.

Findings

Generalization of de Branges theorem to n-variable settings

Characterization of invariant subspaces in multivariable Hardy spaces

New insights into the structure of contractively embedded subspaces

Abstract

This paper focuses on representations of contractively embedded invariant subspaces in several variables. We present a version of the de Branges theorem for $n$-tuples of multiplication operators by the coordinate functions on analytic reproducing kernel Hilbert spaces over the unit ball $\mathbb{B}^n$ and the Hardy space over the unit polydics $\mathbb{D}^n$ in $\mathbb{C}^n$.