On the unitary part of isometries in commuting, completely non doubly commuting pairs

TL;DR

This paper investigates the structure of pairs of commuting isometries that are completely non doubly commuting, revealing the nature of their unitary parts and the limitations on reducing subspaces.

Contribution

It characterizes the unitary components of isometries within pairs that are completely non doubly commuting, highlighting the absence of nontrivial reducing subspaces.

Findings

No nontrivial subspaces reduce both isometries

The unitary part of an isometry in such pairs is uniquely determined

Pairs exhibit specific structural properties related to their non doubly commuting nature

Abstract

There are considered isometries on a Hilbert space. By the Wold theorem any isometry can be decomposed into a unitary operator and a unilateral shift. For a pair of isometries, even commuting, a maximal subspace reducing one isometry to a unitary operator might not reduce the other isometry. In the paper are considered pairs of commuting isometries which are completely non doubly commuting. For such pairs there are no nontrivial subspaces reducing both isometries and one of them to a unitary operator. The results describe a unitary part of an isometry in such a pair.