Pith. sign in

REVIEW 2 major objections 5 minor 23 references

Maximally dissipative and self-adjoint extensions of $K$-invariant operators

T0 review · 2 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that a nonnegative symmetric operator's K-invariance—the identity K*SK=S for a bounded, boundedly invertible K—is automatically inherited by its Friedrichs and Krein–von Neumann extensions.

desk verdict The abstract K-invariance results are solid and worth engaging; the Sturm-Liouville application has a real but fixable coefficient-conditions error. read the letter →

arxiv 2509.05178 v1 pith:YNGHH3CU submitted 2025-09-05 math.SP math.FA

Christoph Fischbacher , Bart Rosenzweig , Jonathan Stanfill This is my paper · ORCID
classification math.SPmath.FA MSC 34B2447A0547B2547A1047B4447B65
keywords K-invariantoperatorsFriedrichsextensionKrein–vonNeumannmaximallydissipativeextensionsSturm–LiouvilleSchröder’sequationJulia’sself-adjoint
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces a notion called K-invariance for operators on Hilbert space: a bounded, boundedly invertible operator K leaves S invariant when K*SK=S. The central theorem is that if a nonnegative symmetric operator S has this property, then its Friedrichs extension and its Krein–von Neumann extension automatically have the same property. It then gives an exact condition, phrased in the Birman–Krein–Vishik–Grubb parametrization, for any other self-adjoint or maximally dissipative extension to be K-invariant. In the Sturm–Liouville setting, with K acting by a weighted change of variable, the invariance conditions reduce to classical functional equations such as Schröder's equation, yielding explicit invariant operators and a full classification of invariant boundary conditions. The interest is that K need not be unitary, so the symmetry is a genuine similarity, and the two canonical extremal extensions are robust under it.

What carries the argument

The central object is the weighted composition operator K, defined in the Sturm–Liouville section as (Kf)(x)=A(x)f(phi(x)); in the abstract section it is simply an arbitrary bounded operator with bounded inverse satisfying K*SK=S. What does the work: (i) the graph-limit description of the Friedrichs and Krein–von Neumann extensions, since K and K^{-1} preserve both the domain and the energy form when S is K-invariant; (ii) the Birman–Krein–Vishik–Grubb parametrization S_B of all nonnegative self-adjoint and maximally dissipative extensions through an auxiliary operator B on ker(S*), where K-invariance is equivalent to D(B)=D(K*BK) and P_{D(B)}K*BK|_{D(B)}=B; and (iii) in the Sturm–Liouville

What would settle it

Test Theorem 2.18 on Example 2.8: with K a scaling and S the Dirichlet Laplacian on (0,infinity), the theorem says only the Dirichlet (Friedrichs) and Neumann (Krein–von Neumann) extensions are K-invariant. Checking that a Robin condition mu in (0,infinity) cannot be invariant—since (Kf)'(0)=lambda^{1/2}f'(0) and (Kf)(0)=lambda^{-1/2}f(0) cannot match mu(Kf)(0) when lambda is not 1—is a direct verification; a failure would disprove the classification.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 2.7: every nonnegative symmetric operator S with K-invariance K*SK=S has K-invariant Friedrichs and Krein–von Neumann extensions. The proof uses the graph-limit characterizations of these two extremal extensions: an approximating sequence from D(S) that is Cauchy in the energy form stays of the same type under left multiplication by K or K^{-1}, because K*SK=S forces both K and K^{-1} to preserve D(S) and the form. The paper goes on to characterize all K-invariant extensions S_B in the Birman–Krein–Vishik–Grubb picture by a condition on the auxiliary operator B, and it specializes this to the finite-defect case, obtaining that root vectors o

Load-bearing premise

For the Sturm–Liouville part, everything depends on the sufficient coefficient conditions in Hypothesis 3.4 actually making the minimal operator K-invariant; the abstract extension theorems themselves only assume K is bounded and boundedly invertible.

Editorial extensions

If this is right

  • Any nonnegative K-invariant symmetric operator automatically has K-invariant Friedrichs and Krein–von Neumann extensions; this holds for every bounded invertible K, not just unitaries.
  • In the Birman–Krein–Vishik–Grubb parametrization, an extension S_B is K-invariant iff its auxiliary operator B satisfies D(B)=D(K*BK) and P_{D(B)}K*BK|_{D(B)}=B; this condition is checkable in concrete examples.
  • When the defect is finite, a K-invariant extension must have ker B containing all root spaces of K restricted to ker(S*) for eigenvalues of modulus not equal to 1; with one-dimensional defect, either exactly two or all maximally dissipative extensions are K-invariant.
  • Sturm–Liouville operators built from coefficient functions solving the functional equations (3.10) are K-invariant, and the K-invariant self-adjoint extensions are exactly those whose boundary conditions satisfy Theorem 3.8; in particular the Krein–von Neumann extension is characterized in Corollary 3.9.
  • There exist nontrivial K-invariant Schrödinger operators on the half-line, such as the Bessel-type example, with the invariance given by a non-unitary weighted change of variable.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The abstract framework is likely to transfer to boundary-triple parametrizations of extensions, where the condition on the auxiliary operator B becomes a concrete boundary-matrix condition; this would extend the classification beyond the parametrization used here.
  • Because the coefficient conditions reduce to Schröder's and Julia's equations, one can generate whole families of K-invariant Sturm–Liouville operators from a single solution by the power and periodic constructions sketched in the paper; a systematic catalogue of such potentials would be a testable by-product.
  • The result that the Friedrichs extension of a K-invariant sectorial operator is K-invariant suggests the same extremal-extension inheritance may hold for other distinguished extensions in the sectorial case, although the paper does not pursue this.
  • The explicit half-line example indicates that non-unitary K-invariances are not exotic: natural Schrödinger operators with a Bessel-type singularity carry them, and the invariant boundary condition for the Krein–von Neumann extension is exactly computable.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper introduces the notion of K-invariance for a densely defined closable operator S with respect to a bounded, boundedly invertible operator K, defined by K^* S K = S. The main abstract results are: the adjoint and closure of a K-invariant operator are K-invariant; the Friedrichs and Krein–von Neumann extensions of a nonnegative K-invariant symmetric operator are always K-invariant; the Friedrichs extension of a K-invariant sectorial operator is K-invariant; and, for strictly positive operators, the K-invariance of extensions described by the Birman–Krein–Vishik–Grubb parameter B is characterized by a domain condition and a projected commutator condition. The paper then applies this framework to Sturm–Liouville operators with K of the form (Kf)(x)=A(x)f(φ(x)), deriving sufficient conditions on p,q,r, characterizing K-invariant boundary conditions, and giving several worked examples including a Bessel-type Schrödinger operator on the half-line. The abstract theorems are carefully argued. The Sturm–Liouville section contains a load-bearing error in the printed coefficient conditions: the transformation rule for p is inverted, so Theorem 3.5 and the examples relying on Hypothesis 3.4 fail as printed, although the correction is local.

Significance. If the Section 3 coefficient condition is corrected, the paper makes a useful contribution. Theorem 2.7 is a clean and non-obvious result: nonnegative K-invariant symmetric operators have K-invariant Friedrichs and Krein–von Neumann extensions for every bounded invertible K, not only unitary K. The extension to sectorial operators and the characterization in Theorem 2.13 are also valuable. The Sturm–Liouville application connects invariance conditions to Schröder's and Julia's equations and provides explicit, nontrivial examples. A particular strength is that the abstract results are derived from stated definitions without fitted parameters or circular reasoning. However, the printed sufficient conditions in Hypothesis 3.4 are internally inconsistent with Remark 3.6 and with the paper's own examples, and this undermines the Sturm–Liouville portion until fixed.

major comments (2)
  1. [§3, Hypothesis 3.4 (Eq. (3.10)), Eq. (1.1), and Theorem 3.5 (Eq. (3.11))] The transformation rule for p is inverted. Let ψ=φ^{-1}(x). Substituting Kf=A(x)f(φ(x)) and the adjoint (3.6) into K^*τK and comparing the coefficient of f'' gives p(x)=A(ψ)^2 φ'(ψ) p(ψ). The printed condition divides by φ'(ψ). The same error appears in (1.1) and in the displayed expression (3.11), where the coefficient of d/dx inside the outer derivative is printed as p(ψ)A(ψ)^2/φ'(ψ) instead of p(ψ)A(ψ)^2 φ'(ψ). This is not a harmless typo: with r=1, q=0, A=1, φ(x)=x/2, C=1, the printed condition p(x)=2p(2x) admits p(x)=c/x, but direct computation gives K^*τK f = (1/4)τf, so T_min is not K-invariant. The q condition in (3.10) is, by contrast, correct.
  2. [§3, Examples 3.10–3.12 and Theorems 3.5, 3.8] As a consequence of the inverted p-condition, Theorem 3.5 is false as stated, and the examples that claim to verify Hypothesis 3.4 do not: for Example 3.10, substituting p(x)=μx^2, A_c=(1+c)^{1/2}, φ_c(x)=(1+c)x/(1+cx) into the printed p-condition gives μ(1+c)^2 x^2/(1+c-cx)^4, which equals μx^2 only for c=0. The corrected condition p(x)=A^2 φ'(ψ)p(ψ) makes the example satisfy the hypothesis. Thus Theorems 3.8 and Corollary 3.9, and Examples 3.10–3.12, are valid only after correcting Eq. (3.10)/(3.11) as described. The abstract results in Section 2 are independent of this issue and appear sound.
minor comments (5)
  1. [§3, Lemma 3.3, Eq. (3.7)] The boundedness estimate has a constant error: using r(x)=C r(φ^{-1}(x)), the change of variables in (3.7) gives the factor C, not C^{-1}. Since C is fixed, boundedness still follows; the displayed constant M=C^{-1} sup(A^2/φ') should be C sup(A^2/φ') (or any finite upper bound).
  2. [§2, Theorem 2.9 proof] In the display after (2.18), the term ∥Kf_n - Kf_n∥^2_A should read ∥Kf_n - Kf_m∥^2_A.
  3. [§2, Theorem 2.13 proof, Eq. (2.25)] In the rewritten expression for K^{-1}ψ, the first summand is written as (Kf_0 + ...); it should be (K^{-1}f_0 + ...), as the subsequent sentence confirms.
  4. [§3, Liouville–Green transformation, Eq. (3.35)] The symbols A and B are reused for the transformed interval endpoints, while A is already the coefficient in K. This overloads notation and is confusing; use different letters (e.g., α, β) for the endpoint values.
  5. [§1 and §3] The paper should explicitly note that Eq. (1.1) and Eq. (3.10) need the p-condition corrected to p(x)=A(φ^{-1}(x))^2 φ'(φ^{-1}(x)) p(φ^{-1}(x)); otherwise Remark 3.6 and the examples will continue to contradict the stated hypothesis.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are derived from the defining condition K*SK=S without any fitted parameter or self-citation chain.

full rationale

The core of the paper (Definition 2.1 onward) is a direct extension-theoretic argument. Theorem 2.7 does not assume the invariance of the Friedrichs/Krein extensions; it proves KD(S_F)=D(S_F) and KD(S_K)=D(S_K) using the sequence characterizations in Proposition 2.6 and the assumed K-invariance of S and S*. Lemma 2.5 then supplies the equivalence to K-invariance. Similarly, Theorem 2.13 proves necessity and sufficiency of D(B)=D(K*BK) and P_{D(B)}K*BK|_{D(B)}=B by manipulating the Grubb parametrization; condition (2.20) is a derived condition, not a conclusion renamed as an assumption. The Sturm-Liouville section states explicit sufficient hypotheses on p, q, r in Hypothesis 3.4 and Theorem 3.5 verifies K*τK=τ by direct computation. Those hypotheses are inputs to a conditional statement, and the examples use them to exhibit operators; no parameter is fitted to the target extension and no extension result is used to define the hypotheses. The author-overlapping references ([8], [9], [11], [12]) cite standard background results (Grubb's parametrization, Sturm-Liouville theory, Krein-von Neumann boundary conditions, Liouville-Green transforms); the key invariance theorems are proved in the paper rather than imported from those citations, so the citations are not load-bearing in the derivation chain. The apparent inconsistency in Eq. (3.10) flagged by a skeptical reading is a correctness issue about sufficient conditions, not a circularity: even if the printed formulas are inverted, the claim would be false rather than true-by-construction. No reduction of a prediction to its input, no uniqueness theorem imported from the authors, and no ansatz smuggled via citation was found.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest entirely on standard extension theory and Sturm-Liouville theory; no free parameters are fitted and no new entities are postulated. The main burden is the correctness of the coefficient conditions in Hypothesis 3.4, which are misprinted in the displayed equations.

assumptions (6)
  • standard math Grubb parametrization of all nonnegative self-adjoint and maximally dissipative extensions of a strictly positive closed symmetric operator (Proposition 2.12)
    Invoked in Theorem 2.13 and cited to Grubb [14] and Fischbacher [8, Thm. 7.2.4].
  • standard math Freudenthal / Ando-Nishio characterization of the Friedrichs and Krein-von Neumann extension domains (Proposition 2.6)
    Used as the foundation of Theorem 2.7; cited to Freudenthal [10], Ando-Nishio [1], and Ashbaugh et al. [4].
  • standard math Kato sectorial form theory, including the formula A_F^* = Q(A) intersect D(A^*)
    Used in Theorem 2.9; from Kato [17, VI section 3] and Arlinskii [2].
  • standard math Parameterization of self-adjoint Sturm-Liouville extensions by separated and coupled boundary conditions (Theorem 3.7)
    Used for Theorem 3.8; cited to Gesztesy-Nichols-Zinchenko [13], Weidmann [22], and Zettl [23].
  • standard math Krein-von Neumann extension of a strictly positive quasi-regular Sturm-Liouville minimal operator is given by coupled boundary conditions
    Used in Corollary 3.9 and in the examples; cited to Fucci-Gesztesy-Kirsten-Littlejohn-Nichols-Stanfill [11, Thm. 3.5].
  • domain assumption Hypothesis 3.4 coefficient conditions are sufficient for K-invariance of the Sturm-Liouville operator
    This is the main application hypothesis. As displayed in (3.10) it contains an internal factor error, so the intended corrected version from Remark 3.6 must be used for the examples to verify the hypothesis.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Maximally dissipative and self-adjoint extensions of $K$-invariant operators." pith.science (2026). https://pith.science/paper/YNGHH3CU

@misc{pith2026250905178,
  author       = {Pith},
  title        = {Pith review of: Maximally dissipative and self-adjoint extensions of $K$-invariant operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YNGHH3CU}},
  note         = {Machine review of arXiv:2509.05178}
}
abstract

We introduce the notion of $K$-invariant operators, $S$, (in a Hilbert space) with respect to a bounded and boundedly invertible operator $K$ defined via $K^*SK=S$. Conditions such that self-adjoint and maximally dissipative extensions of $K$-invariant symmetric operators are also $K$-invariant are investigated. In particular, the Friedrichs and Krein--von Neumann extensions of a nonnegative $K$-invariant symmetric operator are shown to always be $K$-invariant, while the Friedrichs extension of a $K$-invariant sectorial operator is as well. We apply our results to the case of Sturm--Liouville operators where $K$ is given by $(Kf)(x)=A(x)f(\phi(x))$ under appropriate assumptions. Sufficient conditions on the coefficient functions for $K$-invariance to hold are shown to be related to Schr\"oder's equation and all $K$-invariant self-adjoint extensions are characterized. Explicit examples are discussed including a Bessel-type Schr\"odinger operator satisfying a nontrivial $K$-invariance on the half-line.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 21 canonical work pages

  1. [1]

    Ando and K

    T. Ando and K. Nishio,Positive self-adjoint extensions of positive symmetric operators, Tohoku Math. J.22(2), 65–75 (1970)

    work page 1970

  2. [2]

    Arlinski ˘ ı,Extremal extensions of sectorial linear relations, Matematichnii Studii7, 81–96 (1997)

    Yu. Arlinski ˘ ı,Extremal extensions of sectorial linear relations, Matematichnii Studii7, 81–96 (1997)

    work page 1997

  3. [3]

    Aschenbrenner and W

    M. Aschenbrenner and W. Bergweiler,Julia’s equation and differential transcendence, Illinois J. Math.59(2), 277–294 (2015)

    work page 2015

  4. [4]

    M. S. Ashbaugh, F. Gesztesy, M. Mitrea, R. Shterenberg, and G. Teschl,The Krein–von Neumann extension and its connection to an abstract buckling problem, Math. Nachr.283, 165–179 (2010)

    work page 2010

  5. [5]

    Bekker,On non-densely defined invariant Hermitian contractions, Methods Funct

    M. Bekker,On non-densely defined invariant Hermitian contractions, Methods Funct. Anal. Top.13, 223–235 (2007)

    work page 2007

  6. [6]

    Bekker, M

    M. Bekker, M. Bohner, M. Nudel’man, and H. Voulov,Scale-invariant self-adjoint extensions of scale-invariant symmetric operators: continuous versus discrete, Meth. Funct. Anal. Top.21, 41–55 (2015)

    work page 2015

  7. [7]

    Bekker, M

    M. Bekker, M. Bohner, A. Ugol’nikov, and H. Voulov,Parametrization of scale-invariant self-adjoint extensions of scale-invariant symmetric operators, Meth. Funct. Anal. Top.24, 1–15 (2018)

    work page 2018

  8. [8]

    Fischbacher,On the Theory of Dissipative Extensions, PhD Thesis, University of Kent, 2017

    C. Fischbacher,On the Theory of Dissipative Extensions, PhD Thesis, University of Kent, 2017

    work page 2017

Show all 23 references
  1. [9]

    Fischbacher and J

    C. Fischbacher and J. Stanfill,Nonnegative extensions of Sturm–Liouville operators with an application to problems with symmetric coefficient functions, J. Diff. Eq.427, 560–600 (2025)

    2025

  2. [10]

    Freudenthal, ¨Uber die Friedrichssche Fortsetzung halbbeschr¨ ankter Hermitescher Operatoren, Kon

    H. Freudenthal, ¨Uber die Friedrichssche Fortsetzung halbbeschr¨ ankter Hermitescher Operatoren, Kon. Akad. Wten- sch., Amsterdam, Proc.39, 832–633 (1936)

    1936

  3. [11]

    Fucci, F

    G. Fucci, F. Gesztesy, K. Kirsten, L. L. Littlejohn, R. Nichols, and J. Stanfill,The Krein–von Neumann extension revisited, Applicable Anal.101(5), 1593–1616 (2022)

    2022

  4. [12]

    Fucci, M

    G. Fucci, M. Piorkowski, and J. Stanfill,The spectralζ-function for quasi-regular Sturm–Liouville operators, Lett. Math. Phys.115, 8, 48 pp. (2025)

    2025

  5. [13]

    Gesztesy, R

    F. Gesztesy, R. Nichols, and M. Zinchenko,Sturm–Liouville Operators, Their Spectral Theory, and Some Applica- tions, Colloquium Publications, AMS,67, 2024

    2024

  6. [14]

    G. Grubb,A characterization of the non-local boundary value problems associated with an elliptic operator, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3 e s´ erie,22, 425–513 (1968)

    1968

  7. [15]

    Hassi and S

    S. Hassi and S. Kuzhel,On symmetries in the theory of finite rank singular perturbations, J. Funct. Anal.256, 777–809 (2009)

    2009

  8. [16]

    Ibort, F

    A. Ibort, F. Lled´ o, and J. M. P´ erez-Pardo,On Self-Adjoint Extensions and Symmetries in Quantum Mechanics, Ann. Henri Poincar´ e16, 2376–2397 (2015)

    2015

  9. [17]

    Kato,Perturbation Theory for Linear Operators, Second Edition, Springer-Verlag, Berlin Heidelberg New York, 1976

    T. Kato,Perturbation Theory for Linear Operators, Second Edition, Springer-Verlag, Berlin Heidelberg New York, 1976

    1976

  10. [18]

    M. G. Krein,The theory of self-adjoint extensions of semibounded Hermitian transformations and its applications I, Mat. Sbornik20, no. 3, 431–495 (1947)

    1947

  11. [19]

    K. A. Makarov and E. Tsekanovskii,Onµ-scale invariant operators,Meth. Funct. Anal. Top.13, 181–186 (2007)

    2007

  12. [20]

    Schr¨ oder,Ueber iterirte Functionen

    E. Schr¨ oder,Ueber iterirte Functionen. Math. Ann.3, 296–322 (1870)

  13. [21]

    Weidmann,Lineare Operatoren in Hilbertr¨ aumen, Teil I Grundlagen, Verlag B.G

    J. Weidmann,Lineare Operatoren in Hilbertr¨ aumen, Teil I Grundlagen, Verlag B.G. Teubner, Stuttgart/Leipzig/Wiesbaden, 2000 (German)

    2000

  14. [22]

    Weidmann,Lineare Operatoren in Hilbertr¨ aumen

    J. Weidmann,Lineare Operatoren in Hilbertr¨ aumen. Teil II: Anwendungen, Teubner, Verlag B.G. Teubner, Stuttgart/Leipzig/Wiesbaden, 2003 (German)

    2003

  15. [23]

    Zettl,Sturm–Liouville Theory, Mathematical Surveys and Monographs, Vol

    A. Zettl,Sturm–Liouville Theory, Mathematical Surveys and Monographs, Vol. 121, Amer. Math. Soc., Providence, RI, 2005. (C. Fischbacher)Department of Mathematics, Baylor University, Sid Richardson Bldg, 1410 S. 4th Street, W aco, TX 76706, USA Email address:c_fischbacher@baylo...

    2005

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.