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

Abstract: 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}.

• The paper demonstrates that the normalized 3-Brownian shift exhibits the C₀₀-property despite not being power-bounded.
• It provides a full unitary equivalence classification of lifted-type invariant subspaces using spectral and multiplicity data.
• The study extends classical Brownian shift theory to a multivariable setting, offering new insights into m-isometries and operator perturbations.

Invariant Subspaces and the $C_{00}$-Property of $3$-Brownian Shifts: A Technical Overview

Introduction and Motivation

This work advances the analytic and operator-theoretic understanding of Brownian shifts by introducing and analyzing the $3$-Brownian shift $T_{\sigma, \theta}$ on the Hilbert space $$H^2(\mathbb{D}^2) \oplus H^2(\mathbb{D}) \oplus \mathbb{C}$$. This operator extends the classical Brownian shift $B_{\sigma, \theta}$ on $H^2(\mathbb{D}) \oplus \mathbb{C}$ to the setting of $3$-isometries, motivated by recent developments in non-commutative operator theory and the structural analysis of $m$-isometries. The investigation focuses on invariant subspaces of $T_{\sigma, \theta}$, unitary equivalence on lifted-type invariant subspaces, and the asymptotic $3$0-property after normalization.

Structure and Construction of $3$1-Brownian Shifts

The operator $3$2 is defined by the block matrix: $3$3 where $3$4 is multiplication by $3$5 on $3$6, $3$7 is the unilateral shift on $3$8, $3$9 is the canonical embedding from $3$0 to $3$1, and $3$2, $3$3. This form generalizes Brownian unitaries appearing in the recent literature on $3$4-isometries, manifesting as an infinite-rank (non-contractive) perturbation of the shift.

Invariant Subspace Structure

Lifted Type Invariant Subspaces

Analysis reveals that invariant subspaces of $3$5 project onto invariant subspaces of the lower $3$6 Brownian shift $3$7 on $3$8. By exploiting the known classification of Brownian shift invariant subspaces (Type I and Type II per Agler--Stankus), the corresponding lifted invariant subspaces for $3$9 are explicitly described:

• Lifted Type I: If the $T_{\sigma, \theta}$0-component $T_{\sigma, \theta}$1, for inner $T_{\sigma, \theta}$2, then $T_{\sigma, \theta}$3 consists of two summands: functions of the form $T_{\sigma, \theta}$4 and the range of an operator-valued inner function applied to a vector-valued Hardy space over an auxiliary Hilbert space.
• Lifted Type II: If $T_{\sigma, \theta}$5 is of Agler–Stankus Type II, associated with an explicit $T_{\sigma, \theta}$6, then $T_{\sigma, \theta}$7 consists of $T_{\sigma, \theta}$8, $T_{\sigma, \theta}$9, and the range of an operator-valued inner function mapping into the orthogonal complement of $$H^2(\mathbb{D}^2) \oplus H^2(\mathbb{D}) \oplus \mathbb{C}$$0 within $$H^2(\mathbb{D}^2) \oplus H^2(\mathbb{D}) \oplus \mathbb{C}$$1.

Importantly, this structural characterization depends strictly on the Brownian shift subspace type and the nature of operator-valued inner multipliers, extending the Beurling–Lax–Halmos theorem to this $$H^2(\mathbb{D}^2) \oplus H^2(\mathbb{D}) \oplus \mathbb{C}$$2-isometric setting.

Non-Cyclicity and General Subspaces

The paper demonstrates that not all invariant subspaces of $$H^2(\mathbb{D}^2) \oplus H^2(\mathbb{D}) \oplus \mathbb{C}$$3 conform to the lifted type: e.g., $$H^2(\mathbb{D}^2) \oplus H^2(\mathbb{D}) \oplus \mathbb{C}$$4 is not cyclic, and cyclic subspaces generated by certain vectors do not fit the classification, highlighting the complexity of the lattice of invariant subspaces in this higher-dimensional situation.

Unitary Equivalence Classification

A main theorem establishes a full unitary equivalence classification of (nontrivial) lifted Type I and Type II invariant subspaces under $$H^2(\mathbb{D}^2) \oplus H^2(\mathbb{D}) \oplus \mathbb{C}$$5, up to restriction:

• Necessary and sufficient conditions: Restricted $$H^2(\mathbb{D}^2) \oplus H^2(\mathbb{D}) \oplus \mathbb{C}$$6-Brownian shifts are unitarily equivalent if and only if the covariance parameters $$H^2(\mathbb{D}^2) \oplus H^2(\mathbb{D}) \oplus \mathbb{C}$$7 are equal, the auxiliary Hilbert spaces have equal dimension, and in the Type II case, the parameters $$H^2(\mathbb{D}^2) \oplus H^2(\mathbb{D}) \oplus \mathbb{C}$$8 and the $$H^2(\mathbb{D}^2) \oplus H^2(\mathbb{D}) \oplus \mathbb{C}$$9-norms of the corresponding $B_{\sigma, \theta}$0 functions match.

The proof leverages diagonalization arguments, commutant structure, and the intertwining property of unitary maps, extending classical shift equivalence techniques to this more intricate block operator setting. It is established that the equivalence class of the restriction depends tightly on the spectral data and multiplier dimensions.

Power Boundedness and the $B_{\sigma, \theta}$1-Property

The $B_{\sigma, \theta}$2-Brownian shift $B_{\sigma, \theta}$3 is shown not to be power-bounded, as the norm of $B_{\sigma, \theta}$4 exhibits at least linear growth in $B_{\sigma, \theta}$5, precluding similarity to a contraction. However, after normalization by its norm $B_{\sigma, \theta}$6, the sequence of powers of the normalized operator converges strongly to zero in both the operator and adjoint directions, so $B_{\sigma, \theta}$7.

Notably, unlike the classical case, the perturbation from the isometric part is of infinite rank due to the presence of $B_{\sigma, \theta}$8, emphasizing the subspace structure's increased complexity and the subtlety of the asymptotic classification in this regime.

Implications and Future Directions

The extension of Brownian shift theory to $B_{\sigma, \theta}$9-Brownian shifts introduces new challenges and phenomena in multivariable operator theory and the study of $H^2(\mathbb{D}) \oplus \mathbb{C}$0-isometries. This framework aids the understanding of invariant subspace lattices for block-structured, non-normal operators and establishes a pathway for generalizing shift-invariant subspace theory in both commutative and non-commutative directions. The intricate dependence of unitary equivalence on spectral and multiplicity data suggests future work in classifying higher-order Brownian shifts, analyzing multivariate perturbations, and exploring connections to non-commutative function theory and representation theory for operator algebras.

Conclusion

This paper defines and exhaustively studies the $H^2(\mathbb{D}) \oplus \mathbb{C}$1-Brownian shift on $H^2(\mathbb{D}) \oplus \mathbb{C}$2, provides a complete description of its lifted-type invariant subspaces, and characterizes their unitary equivalence classes, demonstrating analytic analogues to classical results and non-trivial extensions in the higher-dimensional setting. The demonstration that the normalized $H^2(\mathbb{D}) \oplus \mathbb{C}$3-Brownian shift belongs to $H^2(\mathbb{D}) \oplus \mathbb{C}$4, despite not being power-bounded, mirrors and extends known results for the classical Brownian shift and provides a foundation for future analysis of $H^2(\mathbb{D}) \oplus \mathbb{C}$5-isometries and their perturbations.

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Reference: "Invariant Subspaces and the $H^2(\mathbb{D}) \oplus \mathbb{C}$6-Property of $H^2(\mathbb{D}) \oplus \mathbb{C}$7-Brownian Shifts" (2604.23447)