Characterization of Invariant subspaces in the polydisc

TL;DR

This paper provides a complete characterization of invariant subspaces for multiplication operators on the Hardy space over the polydisc, solving a longstanding open problem and classifying a broad class of commuting isometries.

Contribution

It offers the first complete description of invariant subspaces in several variables, establishing unitary invariants and classifying commuting isometries on Hardy spaces over the polydisc.

Findings

Complete invariant subspace characterization for $H^2(\

Classification of a large class of commuting isometries.

Results extend to vector-valued Hardy spaces.

Abstract

We give a complete characterization of invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$ on the Hardy space $H^2(\mathbb{D}^n)$ over the unit polydisc $\mathbb{D}^n$ in $\mathbb{C}^n$, $n >1$. In particular, this yields a complete set of unitary invariants for invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$ on $H^2(\mathbb{D}^n)$, $n > 1$. As a consequence, we classify a large class of $n$-tuples, $n > 1$, of commuting isometries. All of our results hold for vector-valued Hardy spaces over $\mathbb{D}^n$, $n > 1$. Our invariant subspace theorem solves the well-known open problem on characterizations of invariant subspaces of the Hardy space over the unit polydisc.