Invariant Subspaces and the $C_{00}$-Property of $3$-Brownian Shifts
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| Format: | Preprint |
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2026
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| _version_ | 1866915958794223616 |
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| author | Nailwal, Rajkamal |
| author_facet | Nailwal, Rajkamal |
| contents | In this paper, we introduce a $3$-Brownian shift $T_{σ, θ}$ 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 $B_{σ, θ}$ 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.$ Here, $M_1$ turns out to be an invariant subspace of the respective Brownian shift $B_{σ, θ}$. 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}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_23447 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Invariant Subspaces and the $C_{00}$-Property of $3$-Brownian Shifts Nailwal, Rajkamal Functional Analysis 30H10, 30J05, 47A15, 47A20, 60J65 In this paper, we introduce a $3$-Brownian shift $T_{σ, θ}$ 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 $B_{σ, θ}$ 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.$ Here, $M_1$ turns out to be an invariant subspace of the respective Brownian shift $B_{σ, θ}$. 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}. |
| title | Invariant Subspaces and the $C_{00}$-Property of $3$-Brownian Shifts |
| topic | Functional Analysis 30H10, 30J05, 47A15, 47A20, 60J65 |
| url | https://arxiv.org/abs/2604.23447 |