Nearly invariant subspaces in the real Hardy space

Abstract: The objective of this article is to study nearly invariant subspaces of the backward shift operator on the real Hardy space. We also investigate nearly invariant subspaces with finite defect, and as a consequence, provide a characterization for almost invariant subspaces of the backward shift operator.

• The paper presents a full structural description for nearly invariant subspaces under the backward shift in the real Hardy space, identifying a selector function g that ensures the real structure.
• It employs complexification and symmetrization techniques to extend Beurling’s theorem, providing clear characterizations and explicit decompositions for finite defect subspaces.
• The work also classifies almost invariant subspaces and delivers precise norm formulas, offering new insights for spectral theory and applications in real-valued analytic signal models.

Nearly Invariant Subspaces in the Real Hardy Space: A Technical Analysis

Introduction and Problem Context

This work undertakes a rigorous and systematic exploration of nearly invariant subspaces of the backward shift operator on the real Hardy space $H^2_\mathbb{R}(\mathbb{D})$. While the nearly invariant subspaces for the complex Hardy space $H^2(\mathbb{D})$ have enjoyed extensive study—rooted in Hitt’s 1988 formulation and Sarason’s functional model—the extension of these concepts to the real Hardy space has remained unexplored until now. The real Hardy space consists of functions in $H^2(\mathbb{D})$ whose Fourier coefficients are real, imparting a distinct algebraic and operator-theoretic structure due to the restriction to the real field.

This paper provides new characterizations for nearly invariant subspaces under the backward shift, analyzes the subclass with finite defect, and, as a corollary, yields a direct classification of almost invariant subspaces for the backward shift on $H^2_\mathbb{R}(\mathbb{D})$ (2604.10240).

Preliminaries: Structure of the Real Hardy Space

The real Hardy space $H^2_\mathbb{R}(\mathbb{D})$ is realized as the closed, real-subspace of $H^2(\mathbb{D})$ comprised of power series with real coefficients. Operator-theoretic considerations revolve around the shift operator $T_z(f)(z) = z f(z)$ and its adjoint, the backward shift $T^*_z(f)(z) = (f(z) - f(0))/z$. The core innovation arises from recasting these operators and the associated notions of (near) invariance in the context of the real—rather than complex—scalar field.

The paper establishes a precise machinery for relating real and complex Hardy spaces via symmetrization and complexification maps, which are subsequently leveraged throughout the main results.

Characterization of Invariant Subspaces

A preliminary result extends Beurling's celebrated theorem to the real Hardy space context, showing that any nonzero shift-invariant subspace of $H^2_\mathbb{R}(\mathbb{D})$ is the range of multiplication by a real inner function: specifically, $\mathcal{M} = \theta H^2_\mathbb{R}(\mathbb{D})$ where $H^2(\mathbb{D})$0 is an inner function with real Fourier coefficients. The proof utilizes the complexification technique and a careful analysis of the conjugation invariance properties of inner functions, facilitating the translation of Beurling’s model into the real setting.

Main Results: Nearly Invariant and Defect Subspaces

Nearly Invariant Subspaces

The central theorem provides a full structural description for nearly invariant subspaces of the real Hardy space under the backward shift:

• If $H^2(\mathbb{D})$1 is a non-trivial subspace nearly invariant under $H^2(\mathbb{D})$2, then there exist a $H^2(\mathbb{D})$3-invariant subspace $H^2(\mathbb{D})$4 of $H^2(\mathbb{D})$5 and a function $H^2(\mathbb{D})$6, orthogonal to $H^2(\mathbb{D})$7 with $H^2(\mathbb{D})$8, such that $H^2(\mathbb{D})$9 and $H^2(\mathbb{D})$0 for all $H^2(\mathbb{D})$1.

The proof is accomplished by constructing the complexification of the real subspace and importing the known Hitt/Sarason theorem. Further, symmetrization and conjugation arguments ensure the requisite real structure is preserved in the representation. A notable point is the verification that the selector $H^2(\mathbb{D})$2 can always be taken to satisfy $H^2(\mathbb{D})$3 (self-conjugacy under the anti-linear involution), which is central to the real structure.

Nearly Invariant Subspaces with Finite Defect

The analysis is extended to nearly invariant subspaces having finite defect, meaning the near invariance is relaxed to allow "escape" into a finite-dimensional defect space. For defect-$H^2(\mathbb{D})$4 nearly invariant subspaces, the paper supplies an explicit decomposition:

• Case (i): If some $H^2(\mathbb{D})$5 satisfies $H^2(\mathbb{D})$6, then $H^2(\mathbb{D})$7 is unitarily equivalent to all vectors of the form $H^2(\mathbb{D})$8, with $H^2(\mathbb{D})$9 belonging to a $H^2_\mathbb{R}(\mathbb{D})$0-invariant real subspace, $H^2_\mathbb{R}(\mathbb{D})$1 as in the previous theorem, and $H^2_\mathbb{R}(\mathbb{D})$2 an orthonormal basis for the defect.
• Case (ii): If every $H^2_\mathbb{R}(\mathbb{D})$3 vanishes at the origin, then $H^2_\mathbb{R}(\mathbb{D})$4 consists of vectors $H^2_\mathbb{R}(\mathbb{D})$5 with $H^2_\mathbb{R}(\mathbb{D})$6 in a $H^2_\mathbb{R}(\mathbb{D})$7-invariant subspace.

The norms of such decomposable functions satisfy precise orthogonality relations, providing strong numerically explicit structure for the subspaces.

Classification of Almost Invariant Subspaces

Building on these results, the paper characterizes almost invariant subspaces (where the backward shift of the entire subspace is contained in itself plus a finite-dimensional space) as precisely those described in the structure theorem for nearly invariant subspaces with finite defect, with a mild additional constraint on $H^2_\mathbb{R}(\mathbb{D})$8 in the non-vanishing case.

Implications and Theoretical Significance

The results close a notable gap in the operator theory of real Hardy spaces by transferring techniques and representation theorems from the classical complex Hardy setting and establishing their validity—and structural adaptations—over the real numbers. The precise identification of the selector function $H^2_\mathbb{R}(\mathbb{D})$9, the role of symmetrization and conjugate invariance, and the interplay between defect spaces and nearly invariant structures are all elucidated in a technically robust manner.

Given the foundational importance of Hardy space submodules in both pure and applied analysis (model theory, spectral theory, systems theory, and function theory), this extension broadens the toolkit for addressing real-valued analytic signal models and potential applications where real symmetry or constraints are manifest.

The explicit norm formulas and the characterizing decompositions for finite defect nearly invariant subspaces may lead to advances in the spectral theory of real shift-type operators, and could inform system identification and real-symmetric functional calculus for operator tuples.

Future Directions

Potential developments inspired by this work include:

• Extension to matrix-valued and vector-valued real Hardy spaces, particularly regarding operator-valued inner functions with real coefficients.
• Investigation of analogous invariance models in other real function spaces, including (real) Dirichlet spaces and associated weighted Hardy spaces.
• Potential connections with real structures in model spaces, de Branges–Rovnyak spaces over the real field, and connections to phase retrieval problems constrained to the real line.

Conclusion

This paper delivers a comprehensive theory of nearly invariant and almost invariant subspaces for the backward shift on the real Hardy space, aligning and extending the rich landscape established for the complex case. By providing concrete structural representations and explicit norm formulas, the work enhances the analytic and algebraic understanding of function-theoretic operator invariance in real Hilbert spaces (2604.10240).