Invariant Subspaces and the $C_{00}$-Property of $3$-Brownian Shifts

Abstract

In this paper, we introduce a $3$-Brownian shift $T_{\sigma, \theta}$ on the Hilbert space $H^2(\mathbb D^2)\oplus H^2(\mathbb D)\oplus \mathbb C,$ which is a natural extension of the classical Brownian shift $B_{\sigma, \theta}$ on $H^2(\mathbb D)\oplus \mathbb C$. This is motivated by Brownian extensions in the context of 3-isometries recently developed by A. Cr\u{a}ciunescu and L. Suciu. We investigate the problem of unitary equivalence for $3$-Brownian shifts on invariant subspaces of the type $M_0 \oplus M_1,$ where $ M_0 \subseteq H^2(\mathbb D^2)$ and $ M_1 \subseteq H^2(\mathbb D)\oplus \mathbb C.$ Here, $M_1$ turns out to be an invariant subspace of the respective Brownian shift $B_{\sigma, \theta}$. We also study the asymptotic behaviour of the normalized $3$-Brownian shifts. This work is motivated by Richter \cite{R88} and very recently by work on Brownian shift on $H^2(\mathbb D)\oplus \mathbb C$ in \cite{DDS2025}.