Canonical Models For Bi-Isometries

TL;DR

This paper introduces a canonical model for bi-isometries, extending the concept from contraction operators, involving analytic operator-valued functions, and explores various conditions and classifications of these models.

Contribution

The paper develops a new canonical model for bi-isometries using contractive analytic functions, expanding the theoretical framework for commuting isometries.

Findings

Introduced a canonical model for bi-isometries.

Classified bi-isometries with shift properties.

Explored pureness conditions and examples.

Abstract

A canonical model, analogous to the one for contraction operators, is introduced for bi-isometries, two commuting isometries on a Hilbert space. This model involves a contractive analytic operator-valued function on the unit disk. Various pureness conditions are considered as well as bi-isometries for which both isometries are shifts. Several families of examples are introduced and classified.