Nearly invariant subspaces in the real Hardy space

Arshad Khan; Dinesh Singh; Sneh Lata

arxiv: 2604.10240 · v1 · submitted 2026-04-11 · 🧮 math.FA

Nearly invariant subspaces in the real Hardy space

Arshad Khan , Sneh Lata , Dinesh Singh This is my paper

Pith reviewed 2026-05-10 15:53 UTC · model grok-4.3

classification 🧮 math.FA

keywords nearly invariant subspacesreal Hardy spacebackward shift operatorfinite defectalmost invariant subspacesinvariant subspacesfunctional analysis

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The pith

Nearly invariant subspaces with finite defect on the real Hardy space characterize almost invariant subspaces for the backward shift.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to study nearly invariant subspaces of the backward shift operator acting on the real Hardy space. It pays special attention to those with finite defect and shows that this leads to a characterization of almost invariant subspaces. A reader would care because invariant subspace problems are central to understanding how operators act on function spaces, and extending this from complex to real cases fills a gap in the theory. The result gives concrete ways to identify subspaces that are nearly preserved by the shift.

Core claim

The authors study nearly invariant subspaces of the backward shift operator on the real Hardy space. They investigate nearly invariant subspaces with finite defect, and as a consequence provide a characterization for almost invariant subspaces of the backward shift operator.

What carries the argument

Nearly invariant subspaces with finite defect, serving to characterize almost invariant subspaces of the backward shift operator.

If this is right

Almost invariant subspaces of the backward shift can be characterized using nearly invariant subspaces with finite defect.
The theory of invariant subspaces in the real Hardy space gains a new tool for classification.
This characterization may help in determining the structure of other related subspaces.
Finite defect provides a quantitative measure for near-invariance under the operator.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This work implies that the real Hardy space may have distinct features from its complex counterpart in terms of subspace invariance.
Explicit constructions of such subspaces could be developed to test the characterization in specific cases.
Potential applications arise in areas like real signal processing where Hardy spaces model certain behaviors.

Load-bearing premise

The notions of nearly invariant subspaces and finite defect extend from the complex Hardy space to the real Hardy space in a way that allows the stated characterization to hold.

What would settle it

A specific subspace of the real Hardy space that is almost invariant under the backward shift but does not arise from a finite defect nearly invariant subspace, or a counterexample where the finite defect condition fails to produce the expected almost invariance, would disprove the characterization.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends nearly invariant subspace results to the real Hardy space and derives a characterization of almost invariant subspaces for the backward shift, but the extension may rest on unverified carry-over of defect-space properties from the complex case.

read the letter

The main takeaway is that this work studies nearly invariant subspaces for the backward shift on the real Hardy space and then uses the finite-defect case to characterize almost invariant subspaces. That characterization is the concrete output the abstract highlights. The extension itself is the part that could matter to people already working on real Hardy spaces, since those spaces appear in contexts where complex conjugation or harmonic extensions matter and the usual complex factorization tools do not apply verbatim. The paper appears to follow the same logical path that has been used in the complex setting: define nearly invariant subspaces, restrict to finite defect, and extract the almost-invariant conclusion as a corollary. That structure is straightforward and keeps the argument short. The soft spot is exactly the one the stress-test note flags. The real Hardy space is typically the closure of real polynomials in L2 or the space of real parts of complex Hardy functions; neither setting automatically inherits the same codimension relations or Beurling-type lemmas that hold over the complex scalars. If the authors only transplant the definitions and cite the complex results without re-checking the inclusion relations or the dimension of the defect space under the real inner product, the implication from finite-defect nearly invariant subspaces to the almost-invariant characterization does not go through. The abstract gives no indication that those steps were redone, so a referee would need to see the actual proofs. This is a niche paper aimed at operator theorists who already care about invariant-subspace questions on Hardy spaces and who want to see the real-variable version written down. It is not broad enough to interest a general audience, but the question it poses is well-posed and the claimed consequence is falsifiable once the proofs are checked. I would send it to peer review so that the real-complex transfer can be examined directly rather than desk-rejecting it on the abstract alone.

Referee Report

1 major / 1 minor

Summary. The manuscript studies nearly invariant subspaces of the backward shift operator on the real Hardy space. It examines the subclass with finite defect and derives from this a characterization of almost invariant subspaces for the backward shift.

Significance. If the central implication holds, the work extends the theory of nearly invariant and almost invariant subspaces from the complex to the real Hardy space, which may be of interest for applications involving real-valued analytic functions. The explicit focus on finite-defect cases and the resulting characterization constitute the main potential contribution, provided the necessary structural equivalences are established rigorously in the real setting.

major comments (1)

[Section on finite-defect nearly invariant subspaces and the subsequent characterization] The central consequence (characterization of almost invariant subspaces) is stated to follow from the study of nearly invariant subspaces with finite defect. The manuscript must explicitly re-derive or verify that the defect space (finite-dimensional complement to S^*M inside M) and the relevant inclusion/codimension relations remain unchanged under the real inner product and the real Hardy-space realization (closure of real polynomials or real harmonic extensions). Without this verification, the implication from finite-defect nearly invariant subspaces to the characterization does not automatically carry over, as factorization and Beurling-type lemmas do not transfer verbatim.

minor comments (1)

[Introduction] Notation for the real Hardy space and the backward shift operator should be introduced with explicit definitions early in the paper to avoid ambiguity with the complex case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive suggestion regarding the transfer of structural properties from the complex to the real Hardy space. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Section on finite-defect nearly invariant subspaces and the subsequent characterization] The central consequence (characterization of almost invariant subspaces) is stated to follow from the study of nearly invariant subspaces with finite defect. The manuscript must explicitly re-derive or verify that the defect space (finite-dimensional complement to S^*M inside M) and the relevant inclusion/codimension relations remain unchanged under the real inner product and the real Hardy-space realization (closure of real polynomials or real harmonic extensions). Without this verification, the implication from finite-defect nearly invariant subspaces to the characterization does not automatically carry over, as factorization and Beurling-type lemmas do not transfer verbatim.

Authors: We agree that an explicit verification is necessary to make the implication rigorous in the real setting. In the revised manuscript we will insert a dedicated paragraph (or short subsection) immediately preceding the characterization theorem. There we verify directly that if M is nearly invariant for the backward shift S^* on the real Hardy space H^2_R, then the defect space M ⊖ S^* M is finite-dimensional precisely when the corresponding complexification satisfies the same property, and that the codimension relations are preserved because the real inner product coincides with the restriction of the complex one and the real Hardy space is the closed linear span of real polynomials (or real harmonic extensions). This allows the finite-defect assumption to imply the same inclusion and codimension statements used in the complex case, so that the characterization of almost invariant subspaces follows without relying on verbatim transfer of factorization lemmas. The added verification uses only the definition of the real Hardy space and the fact that conjugation commutes with S^*. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation appears self-contained

full rationale

The paper's stated objective is to study nearly invariant subspaces of the backward shift on the real Hardy space and derive a characterization of almost invariant subspaces as a consequence of the finite-defect case. No equations, self-citations, fitted parameters, or ansatzes are quoted in the available text that would reduce any claimed prediction or characterization to its own inputs by construction. The extension from complex to real Hardy space is presented as the direct object of study rather than presupposed via unverified equivalence or prior author results. This is the normal case of an independent operator-theoretic investigation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; no explicit free parameters, new axioms, or invented entities are stated.

axioms (1)

standard math Standard properties of the backward shift operator and Hardy spaces from prior functional analysis literature
The abstract invokes these without re-deriving them.

pith-pipeline@v0.9.0 · 5320 in / 1036 out tokens · 19628 ms · 2026-05-10T15:53:28.377057+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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I. Chalendar, N. Chevrot, and J. R. Partington,Nearly invariant subspaces for back- wards shifts on vector-valued Hardy spaces, J. Oper. Theory63(2010), no. 2, 403–

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Chalendar, E

I. Chalendar, E. A. Gallardo-Guti´ errez, and J. R. Partington,A Beurling theorem for almost-invariant subspaces of the shift operator, J. Oper. Theory83(2020), no. 2, 321–331. MR4078702

work page 2020
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Chattopadhyay and S

A. Chattopadhyay and S. Das,Study of nearly invariant subspaces with finite defect in Hilbert spaces, Proc. Indian Acad. Sci. Math. Sci.132(2022), no. 1, Paper No. 10,

work page 2022
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Chattopadhyay, S

A. Chattopadhyay, S. Das, and C. Pradhan,Almost invariant subspaces of the shift operator on vector-valued Hardy spaces, Integral Equ. Oper. Theory92(2020), no. 6, Paper No. 52, 15. MR4181859

work page 2020
[5]

Das and J

S. Das and J. Sarkar,Invariant subspaces of perturbed backward shift, arXiv preprint arXiv:2407.17352 (2024)

work page arXiv 2024
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de Branges and J

L. de Branges and J. Rovnyak,Square summable power series, Courier Corporation, 2015

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C. Erard,Nearly invariant subspaces related to multiplication operators in Hilbert spaces of analytic functions, Integral Equ. Oper. Theory50(2004), no. 2, 197–210. MR2099789

work page 2004
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Jupiter and D

D. Jupiter and D. Redett,Invariant subspaces ofRL 1, Houston J. Math.32(2006), no. 4, 1133–1138. MR2268475

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A. Khan, S. Lata, and D. Singh,Nearly invariant Brangesian subspaces, Canad. Math. Bull.68(2025), no. 1, 318–337. MR4862244

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,Perturbations of Toeplitz operators on vector-valued Hardy spaces, arXiv preprint arXiv:2507.05721 (2025)

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Y. Liang and J. R. Partington,Nearly invariant subspaces for operators in Hilbert spaces, Complex Anal. Oper. Theory15(2021), no. 1, Paper No. 5, 17. MR4179960

work page 2021
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Mehanna and D

A. Mehanna and D. Singh,Invariant subspace of functions inH 1 with real Taylor coefficients, Aligarh Bull. Math.12(1987), 45–50. MR1109774

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Mikkola and A

K. Mikkola and A. Sasane,Tolokonnikov’s lemma for realH ∞ and the real disc algebra, Complex Anal. Oper. Theory1(2007), no. 3, 439–446. 16 ARSHAD KHAN, SNEH LATA, AND DINESH SINGH

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Raghupathi and D

M. Raghupathi and D. Singh,Function theory in real Hardy spaces, Math. Nachr. 284(2011), no. 7, 920–930. MR2815961

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Sarason,Nearly invariant subspaces of the backward shift, Oper

D. Sarason,Nearly invariant subspaces of the backward shift, Oper. Theory: Advances Applicat.35(1988), 481–493. MR1017680

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B. D. Wick,A note about stabilization inA R(D), Math. Nachr.282(2009), no. 6, 912–916

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Mat.54(2010), no

,Stabilization inH ∞ R (D), Publ. Mat.54(2010), no. 1, 25–52. Department Of Mathematics, School of Natural Sciences, Shiv Nadar In- stitution Of Eminence, Gautam Buddha Nagar - 201314, Uttar Pradesh, India Email address:ak954@snu.edu.in Department Of Mathematics, School of Natural Sciences, Shiv Nadar In- stitution Of Eminence, Gautam Buddha Nagar - 20131...

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[1] [1]

Chalendar, N

I. Chalendar, N. Chevrot, and J. R. Partington,Nearly invariant subspaces for back- wards shifts on vector-valued Hardy spaces, J. Oper. Theory63(2010), no. 2, 403–

work page 2010

[2] [2]

Chalendar, E

I. Chalendar, E. A. Gallardo-Guti´ errez, and J. R. Partington,A Beurling theorem for almost-invariant subspaces of the shift operator, J. Oper. Theory83(2020), no. 2, 321–331. MR4078702

work page 2020

[3] [3]

Chattopadhyay and S

A. Chattopadhyay and S. Das,Study of nearly invariant subspaces with finite defect in Hilbert spaces, Proc. Indian Acad. Sci. Math. Sci.132(2022), no. 1, Paper No. 10,

work page 2022

[4] [4]

Chattopadhyay, S

A. Chattopadhyay, S. Das, and C. Pradhan,Almost invariant subspaces of the shift operator on vector-valued Hardy spaces, Integral Equ. Oper. Theory92(2020), no. 6, Paper No. 52, 15. MR4181859

work page 2020

[5] [5]

Das and J

S. Das and J. Sarkar,Invariant subspaces of perturbed backward shift, arXiv preprint arXiv:2407.17352 (2024)

work page arXiv 2024

[6] [6]

de Branges and J

L. de Branges and J. Rovnyak,Square summable power series, Courier Corporation, 2015

work page 2015

[7] [7]

Erard,Nearly invariant subspaces related to multiplication operators in Hilbert spaces of analytic functions, Integral Equ

C. Erard,Nearly invariant subspaces related to multiplication operators in Hilbert spaces of analytic functions, Integral Equ. Oper. Theory50(2004), no. 2, 197–210. MR2099789

work page 2004

[8] [8]

Hayashi,The kernel of a Toeplitz operator, Integral Equ

E. Hayashi,The kernel of a Toeplitz operator, Integral Equ. Oper. Theory9(1986), 588–591

work page 1986

[9] [9]

Hitt,Invariant subspaces ofH 2 of an annulus, Pacific J

D. Hitt,Invariant subspaces ofH 2 of an annulus, Pacific J. Math.134(1988), no. 1, 101–120. MR953502

work page 1988

[10] [10]

Jupiter and D

D. Jupiter and D. Redett,Invariant subspaces ofRL 1, Houston J. Math.32(2006), no. 4, 1133–1138. MR2268475

work page 2006

[11] [11]

A. Khan, S. Lata, and D. Singh,Nearly invariant Brangesian subspaces, Canad. Math. Bull.68(2025), no. 1, 318–337. MR4862244

work page 2025

[12] [12]

,Perturbations of Toeplitz operators on vector-valued Hardy spaces, arXiv preprint arXiv:2507.05721 (2025)

work page arXiv 2025

[13] [13]

Liang and J

Y. Liang and J. R. Partington,Nearly invariant subspaces for operators in Hilbert spaces, Complex Anal. Oper. Theory15(2021), no. 1, Paper No. 5, 17. MR4179960

work page 2021

[14] [14]

Mehanna and D

A. Mehanna and D. Singh,Invariant subspace of functions inH 1 with real Taylor coefficients, Aligarh Bull. Math.12(1987), 45–50. MR1109774

work page 1987

[15] [15]

Mikkola and A

K. Mikkola and A. Sasane,Tolokonnikov’s lemma for realH ∞ and the real disc algebra, Complex Anal. Oper. Theory1(2007), no. 3, 439–446. 16 ARSHAD KHAN, SNEH LATA, AND DINESH SINGH

work page 2007

[16] [16]

Raghupathi and D

M. Raghupathi and D. Singh,Function theory in real Hardy spaces, Math. Nachr. 284(2011), no. 7, 920–930. MR2815961

work page 2011

[17] [17]

Sarason,Nearly invariant subspaces of the backward shift, Oper

D. Sarason,Nearly invariant subspaces of the backward shift, Oper. Theory: Advances Applicat.35(1988), 481–493. MR1017680

work page 1988

[18] [18]

B. D. Wick,A note about stabilization inA R(D), Math. Nachr.282(2009), no. 6, 912–916

work page 2009

[19] [19]

Mat.54(2010), no

,Stabilization inH ∞ R (D), Publ. Mat.54(2010), no. 1, 25–52. Department Of Mathematics, School of Natural Sciences, Shiv Nadar In- stitution Of Eminence, Gautam Buddha Nagar - 201314, Uttar Pradesh, India Email address:ak954@snu.edu.in Department Of Mathematics, School of Natural Sciences, Shiv Nadar In- stitution Of Eminence, Gautam Buddha Nagar - 20131...

work page 2010