Nearly Dual Compressed Shift-Invariant Subspaces

• The paper introduces a generalized framework for nearly dual compressed shift-invariant subspaces via an extension of Hitt’s algorithm, providing explicit decompositions using defect vectors.
• It details a unique splitting of any closed subspace into analytic and co-analytic parts by leveraging the dual compressed shift operator and its near-invariance property.
• The work establishes a comprehensive classification linking classical Hardy space model theory to dual truncated Toeplitz operators and spectral analysis.

A nearly dual compressed shift-invariant subspace refers to a closed subspace of the orthogonal complement of a model space, K₍θ₎^⊥, that is nearly invariant under the dual compressed shift operator. The structure of such subspaces is elucidated via extensions of Hitt’s algorithm, providing a detailed decomposition analogous to the classical theory for model spaces but adapted to the structural idiosyncrasies of the dual compressed shift (Chattopadhyay et al., 14 Oct 2025).

1. Structural Context and Operator Framework

Model spaces K₍θ₎ = H² ⊖ θH², for inner functions θ, are classically backward shift (S*)-invariant and occupy a central role in operator theory on H². Their orthogonal complements K₍θ₎^⊥ in L²(𝕋), which can be realized as θH² ⊕ overline{H₀²} (where H₀² = zH²), serve as domains for dual formulations of compressions of the shift operator.

The dual compressed shift U₍θ₎: K₍θ₎^⊥ → K₍θ₎^⊥ is defined as: $U₍θ₎(h) = (I - P₍θ₎)(z h)$ where P₍θ₎ is the orthogonal projection onto K₍θ₎. Its explicit action, as well as that of its adjoint U₍θ₎*, are computed in terms of the Fourier coefficients and the geometry induced by θ.

2. Notion of Near Invariance for Dual Compressed Shifts

A closed subspace ℳ ⊆ K₍θ₎^⊥ is nearly U₍θ₎-invariant if: $$U₍θ₎\left(\mathcal{M} \cap U₍θ₎^* K₍θ₎^⊥\right) \subseteq \mathcal{M}$$ That is, whenever h ∈ ℳ ∩ U₍θ₎^* K₍θ₎^⊥, one has U₍θ₎(h) ∈ ℳ. Analogous to nearly S*-invariant subspaces (see [Hitt–Sarason]), this property allows for a finite-dimensional "defect" or failure of strict invariance under the dual compressed shift.

The notion is symmetric for the adjoint: nearly U₍θ₎-invariant subspaces are defined similarly via the action of U₍θ₎.

3. Decomposition via Hitt’s Algorithm

Hitt’s algorithm, originally developed to classify nearly S*-invariant subspaces in H², is adapted in this setting to nearly dual compressed shift-invariant subspaces. The key steps are:

• Any closed subspace ℳ ⊆ K₍θ₎^⊥ can be split as

$\mathcal{M} = \mathcal{M}_- \oplus \mathcal{M}_+$

where $\mathcal{M}_-$ is a closed subspace of overline{H₀²} (negative frequency part), and $\mathcal{M}_+$ is a closed subspace of θH² (positive frequency part).

• If ℳ is not entirely orthogonal to 𝑧̄ (the conjugate-linear functional corresponding to the -1st Fourier coefficient is nonzero), then the "defect" Δ(ℳ) = ℳ \ (ℳ ∩ {𝑧̄}^⊥) is one-dimensional and spanned by a unit vector g with (g)₋₁ = ⟨g, 𝑧̄⟩ > 0.
• For any f ∈ ℳ, there is a unique decomposition:

$$f = a_0 g + f_1,\quad f_1 \in \mathcal{M} \cap \{𝑧̄\}^⊥$$

• Iteratively applying a rank-one modified operator

$R_g(h) = U₍θ₎ (I - g \otimes g)h$

peels away the contributions in the direction of the "defect" vector g, yielding the representation:

$f = g \cdot \sum_{k=0}^{\infty} a_k 𝑧̄^k + f_{+,N}$

with the remainder f_{+,N} in the positive-frequency part and the series converging in overline{H²}.

This recursive procedure is effectively Hitt’s algorithm extended to K₍θ₎^⊥, producing a structural theorem for all nearly dual compressed shift-invariant subspaces.

4. Complete Classification of Nearly Dual Compressed Shift-Invariant Subspaces

The main theorem obtained via this decomposition (Theorem 3.7 (Chattopadhyay et al., 14 Oct 2025)) establishes that any nearly U₍θ₎-invariant subspace ℳ ⊆ K₍θ₎^⊥ is one of the following forms:

• θH²
• γ θH² for some (possibly constant) inner function γ with γ(0) ≠ 0
• overline{H₀²}
• g overline{K_α} for some inner α (with suitable normalization of g)
• overline{H₀²} ⊕ γ θH²
• g overline{K_α} ⊕ γ θH²
• g overline{K_α} ⊕ θH²
• overline{η H²} for inner η
• overline{η H²} ⊕ θH²
• overline{η H²} ⊕ γ θH²

where K_α = H² ⊖ αH², and overline{K_α} ⊂ overline{H²}.

This classification encompasses all possibilities for the structure of nearly dual compressed shift-invariant subspaces, characterizing them in terms of inner functions and (when appropriate) a one-dimensional "defect" vector.

5. Interaction with Hardy Space and Model Space Structures

The decomposition makes explicit the "dual" relationship between classical model spaces and their orthogonal complements when analyzed under compression by shift or dual shift operators:

• The analytic (θH²) component encapsulates the behavior parallel to classical S-invariant subspaces in H².
• The co-analytic (overline{H₀²}, overline{K_α}, etc.) component is governed by the almost-invariance under the bilateral or dual compressed shift, distinguishable by the spectral support in the negative frequencies.
• When the defect vanishes (i.e., ℳ ⊆ {𝑧̄}^⊥), the structure reduces to classical invariant subspaces. When the defect is nontrivial, the decomposability into "defect-generated" and invariant parts is essential.

6. Role of Rank-One Corrections and Connection to Dual Truncated Toeplitz Operators

A key feature is the use of rank-one corrections in the recursive decomposition, accounting for the "defect" that prevents strict invariance. This mechanism parallels the role of finite-rank perturbations in the theory of nearly invariant subspaces for Toeplitz and truncated Toeplitz operators. Notably, certain nearly dual compressed shift-invariant subspaces arise as kernels or images of dual truncated Toeplitz operators, emphasizing the algebraic and geometric interplay between operator-theoretic perturbation and function-theoretic structure.

7. Broader Implications and Applications

This structural theory for nearly dual compressed shift-invariant subspaces enables:

• A generalization of the classical decomposition for nearly invariant subspaces (as in Hitt and Sarason) to the dual model context.
• An explicit parametrization of all possible nearly invariant subspaces in the dual setting, preparing the ground for a comprehensive description of invariant subspace lattices for dual compressions and their perturbations.
• Direct applications to the spectral analysis, model theory for contractions, and the structure theory of dual (or multiband) truncated Toeplitz operators.

This synthesis clarifies the foundational role of Hitt’s algorithm in structuring nearly dual compressed shift-invariant subspaces and relates dual model space theory to recent developments in the analysis of perturbations and generalizations of classical invariant subspace results (Chattopadhyay et al., 14 Oct 2025).