The structure of an isometric tuple

TL;DR

This paper generalizes the classical decomposition of an isometry to n-tuples of operators, providing a structural breakdown that influences the understanding of the associated operator algebras.

Contribution

It introduces a decomposition for isometric tuples that extends the Lebesgue-von Neumann-Wold theorem to higher dimensions.

Findings

Decomposition of isometric tuples into shift and unitary parts

Determination of algebraic structures from the decomposition

Extension of classical isometry theory to n-tuples

Abstract

An $n$-tuple of operators $(V_1,...,V_n)$ acting on a Hilbert space $H$ is said to be isometric if the operator $[V_1\...\ V_n]:H^n\to H$ is an isometry. We prove a decomposition for an isometric tuple of operators that generalizes the classical Lebesgue-von Neumann-Wold decomposition of an isometry into the direct sum of a unilateral shift, an absolutely continuous unitary and a singular unitary. We show that, as in the classical case, this decomposition determines the weakly closed algebra and the von Neumann algebra generated by the tuple.