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

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. Crăciunescu 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 $M_1\subseteq H^2(\mathbb{D})\oplus \mathbb{C}\mathrm{.}$ 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}.