Invariant subspaces and the $C_{00}$-property of Brownian Shifts

TL;DR

This paper studies the structure of Brownian shifts on invariant subspaces, classifies their unitary equivalence, and shows that normalized Brownian shifts asymptotically belong to the $C_{00}$ class, revealing new spectral properties.

Contribution

It provides a classification of Brownian shifts restricted to invariant subspaces and establishes their asymptotic $C_{00}$-property, a novel spectral analysis result.

Findings

Classification of Brownian shifts on invariant subspaces.

Proof that normalized Brownian shifts are asymptotically in $C_{00}$.

Identification of conditions for unitary equivalence of Brownian shifts.

Abstract

We consider the restriction of Brownian shifts to their invariant subspaces and classify when they are unitarily equivalent. Additionally, we prove an asymptotic property stating that normalized Brownian shifts belong to the classical $C_{00}$-class.