On extensions of commuting tuples of symmetric and isometric operators

Sergey M. Zagorodnyuk

REVIEW 2 major objections 3 minor 20 references

On extensions of commuting tuples of symmetric and isometric operators

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read A commuting tuple of symmetric and isometric operators admits, under a conjugation symmetry, a self-adjoint extension of its first member that commutes with the rest.

desk verdict Worth a serious look: the multidimensional Godic–Lucenko theorem with a common conjugation is genuinely new, and the extension/moment-problem results hold up despite a few presentation typos. read the letter →

arxiv 1908.00794 v1 pith:KSXJLF3E submitted 2019-08-02 math.FA

Sergey M. Zagorodnyuk This is my paper · ORCID

classification math.FA MSC 47A1347A5747B25

keywords commutingoperatorssymmetricisometricself-adjointextensionsunitaryconjugationpower-trigonometricmomentproblemGodič–Lucenkotheorem

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

This paper asks when a finite family of pairwise commuting operators, some symmetric and some isometric, can be extended to a family of pairwise commuting self-adjoint and unitary operators. It proves that if there is a conjugation symmetry relating the isometries to their inverses and fixing the symmetric operators, then the first symmetric operator has a self-adjoint extension in the same Hilbert space that still commutes with every other member. The proof passes through a multidimensional version of the Godič–Lucenko theorem, which factors any finite family of commuting unitaries using one shared conjugation. An application solves a multidimensional power-trigonometric moment problem, recovering the one-dimensional criterion and adding a boundedness condition in higher dimension.

What carries the argument

The carrying mechanism is the Cayley-transform decomposition of $A_1$ into four subspaces $H_1=\overline{(A_1-z_0)D(A_1)}$, $H_2=H\ominus H_1$, $H_3=\overline{(A_1-\bar z_0)D(A_1)}$, $H_4=H\ominus H_3$, together with the isometric patching operator $U_{2,4}=JK$ that maps $H_2$ onto $H_4$. The extension $\hat{A}_1$ is the inverse Cayley transform of $V_1\oplus U_{2,4}$, where $V_1$ is the Cayley part on $H_1\to H_3$. The conjugation $J$ from condition (d) is exactly what forces $U_{2,4}$ to commute with each $B_k$, and the multidimensional Godič–Lucenko theorem provides a common conjugation $K$ so that the restricted Cayley transforms of the other symmetric operators and the isometries all commute with it, making $\hat{A}_1$ commute with the whole tuple.

What would settle it

A decisive calculation: take $H=L^2([0,1])$, $A_1=-i\,d/dx$ on the domain of absolutely continuous functions with $f(0)=f(1)=0$, and $B$ the unitary multiplication by $e^{2\pi ix}$. These two operators commute, and $B$ is unitary. Verifying whether a conjugation $J$ with $JA_1=A_1J$ and $JBJ=B^{-1}$ exists is a finite computation; if it does, the theorem predicts an explicit self-adjoint extension of $A_1$ commuting with $B$, which can be checked directly. A tuple satisfying (a)–(d) for which no such extension exists would refute the theorem.

Watch

Extended reading notes

Core claim

The central discovery is the extension theorem (Theorem 4). For a commuting tuple $T=(A_1,\ldots,A_\rho,B_1,\ldots,B_\tau)$ with joint invariant dense domain $D$ in a separable Hilbert space $H$, suppose the later symmetric operators are essentially self-adjoint, the isometries are essentially unitary, their closures commute pairwise, $B_kD=D$, a relative essential-self-adjointness condition holds for $A_2,\ldots,A_\rho$ on $(A_1-z_0)D$, and there is a conjugation $J$ with $JD\subseteq D$, $A_jJ=JA_j$ for all $j$, and $B_kJ=JB_k^{-1}$ for all $k$. Then there exists a self-adjoint operator $\hat{A}_1\supseteq A_1$ in $H$ that commutes with every $A_j$ and $B_k$. The paper also proves that any finite family of commuting unitary operators on a separable Hilbert space factors as $U_k=J_kC$ with a single common conjugation $C$, a multidimensional analogue of the Godič–Lucenko theorem that supplies the joint factorization used in the main proof.

Load-bearing premise

The load-bearing premise is condition (d): there exists a conjugation (an antiunitary involution) that preserves the joint domain, commutes with each symmetric operator, and sends each isometry to its inverse; without such a symmetry the patching operator $U_{2,4}=JK$ cannot be proved to commute with the isometries, and the theorem offers no alternative route.

Editorial extensions

If this is right

Under Theorem 4, a self-adjoint extension of $A_1$ exists in the original space $H$, not in a larger extension space, and it commutes with the closures of all other tuple members.
When $A_2,\ldots,A_\rho$ are bounded, condition (c) is automatic, so the same conclusion follows from conditions (b) and (d) together with boundedness (Corollary 1).
Any pairwise commuting finite family of unitaries on a separable Hilbert space admits a joint factorization $U_k=J_kC$ with one common conjugation, extending the single-operator Godič–Lucenko theorem.
For the multidimensional power-trigonometric moment problem, the positive-kernel condition (30) is necessary in every dimension and sufficient in dimension one; in higher dimensions it is sufficient together with the boundedness condition (B) (Theorem 5).

Reading between the lines

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

Condition (d) can be read as a time-reversal symmetry of the tuple: the isometries must be antiunitarily equivalent to their inverses while the symmetric operators are left fixed. Tuples that lack such a reversal symmetry sit outside the theorem even if they satisfy every other condition.
The common-conjugation factorization in Theorem 3 implies that a commuting family of unitaries is simultaneously complex symmetric with respect to one fixed conjugation, a fact the extension proof uses but which may have independent uses in model theory of commuting tuples.
The proof only extends the first symmetric operator; iterating the construction on the resulting tuple would require the conjugation condition to survive after extension, and it is not shown that it does. Whether such iteration is possible is a natural next step.
Condition (B) in Theorem 5 is an artifact of forcing boundedness; a variant of the extension theorem allowing unbounded closures might remove it and make the kernel condition alone sufficient in all dimensions.

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.

Desk Editor's Note

read the letter

The paper is a solid operator-theory contribution. The main new thing is Theorem 3, a multidimensional Godic–Lucenko factorization in which the same conjugation K works for all unitary operators in a commuting tuple. That is a real extension of the classical single-unitary theorem and it is what makes the rest of the paper work. Theorem 4 then uses it to get a commuting self-adjoint extension of A1 that commutes with the other symmetric operators and with the isometric operators, under the stated conditions. The application to a multidimensional Devinatz moment problem (Theorem 5) is reasonable, and the extra boundedness assumption for r≥2 is made explicit.

I read the proof of Theorem 4 carefully, because the reader flagged a possible circularity. It is not circular. The two places where the text says 'apply Theorem 4' to the restricted unitaries are misprints: the intended reference is Theorem 3, and the argument works with that substitution. The degenerate Cayley formula with (z0−z0) is also a typo for the standard Cayley transform. These are real presentational flaws that should be fixed, but they do not affect the math.

The genuinely load-bearing hypothesis is condition (d), the existence of a conjugation J with the given commutation relations. Without it the construction of U_{2,4}=JK collapses and the theorem provides no alternative. That is a substantial limitation, but the theorem states it clearly, so it is not a hidden gap. The paper also relies heavily on the author's prior work ([18] and [19]). That is acceptable because [19] is a model theorem for commuting self-adjoint and unitary operators, and [18] is the two-operator case; neither already contains the tuple extension result.

For whom: operator theorists working on symmetric-operator extensions, commuting tuples, and moment problems. The paper deserves a serious referee. It is not a desk-reject. It needs minor revisions: correct the misprints, clarify the references in the proof of Theorem 4, and maybe add a remark that condition (d) is necessary in a strong sense. I would be comfortable with a conditional accept after those fixes.

Referee Report

2 major / 3 minor

Summary. The paper studies extension problems for commuting tuples of symmetric and isometric operators on a separable Hilbert space. It proves a multidimensional version of the Godič-Lucenko theorem (Theorem 3): every tuple of pairwise commuting unitary operators can be written as U_k = J_k C with a common conjugation C, and also in the dual form U_k = K L_k with a common K. Building on this and the author's prior two-operator result ([18, Theorem 1]), the paper proves Theorem 4, which gives sufficient conditions for A_1 to have a self-adjoint extension commuting with the remaining operators in a tuple of symmetric and isometric operators, and a corollary for bounded operators. These results are then applied to a multidimensional power-trigonometric moment problem (Theorem 5), yielding sufficiency of the natural positivity condition (30) together with a boundedness condition (B) when r ≥ 2, and recovering the classical Devinatz-type result for r = 1.

Significance. If correct, the paper provides a substantial generalization of the author's earlier two-operator extension theorem to arbitrary commuting tuples, including isometric operators, and establishes a multidimensional analog of the Godič-Lucenko factorization theorem that is likely to be of independent interest. The application to the multidimensional power-trigonometric moment problem is nontrivial and gives a concrete sufficient condition for solvability. The proof strategy is coherent: Theorem 3 follows from the author's model theorem [19], and Theorem 4 uses a standard Cayley-transform construction together with the common-conjugation factorization. The main concern is that the printed proof of Theorem 4 contains two glaring typographical errors (self-referential citations to Theorem 4 and an impossible Cayley transform formula) that make the proof appear circular and formally invalid until corrected.

major comments (2)

[§2, Proof of Theorem 4, Eqs. (22)–(23)] The proof of Theorem 4 contains two explicit references to 'apply Theorem 4' in the course of proving Theorem 4 itself: once for the operators B_{k,2} in the ρ=1 case and once for the tuple (U_{j,2}, B_{k,2}) in the ρ≥2 case. Read literally, this is a circular argument. The correct reference is evidently Theorem 3, which provides the required factorization of pairwise commuting unitaries into products of conjugations with a common conjugation. As printed, the logical chain of the main theorem is broken; please correct these citations (and similarly the phrase 'By the Godič-Lucenko Theorem' at Eq. (20), which should refer to Theorem 3).
[§2, Proof of Theorem 4, first displayed equation] The Cayley transform is written as V1 := (A1 - z0 E_H)(A1 - z0 E_H)^{-1} = E_H + (z0 - z0)(A1 - z0 E_H)^{-1}. With the formula as typeset, the right-hand side equals E_H and the transform is the identity, which would make the subsequent decomposition H = H1 ⊕ H2 = H3 ⊕ H4 and the isometric map U_{2,4} meaningless. The intended formula is clearly the standard Cayley transform with the second factor (A1 - \bar z0 E_H)^{-1} and coefficient (z0 - \bar z0). Please correct this and check the manuscript for other missing overlines.

minor comments (3)

[§2, Proof of Theorem 4, final paragraph] The sentence 'If ρ ≥ 2, the considerations after (20) show that \hat A1 commutes with Aj (j ∈ Z2,ρ) as well' is terse. Since the choice of a common K is the crux of the multidimensional part, please add a short explicit verification that the inverse Cayley transform of V1 ⊕ J K commutes with each A_j when K satisfies K U_{j,2} K = U_{j,2}^{-1}.
[Theorem 5, condition (B)] In condition (B), the constants C_j should be explicitly stated to be independent of the finite sequence α_{m,n}; the wording 'for all finite sequences' makes this clear, but spelling it out would remove ambiguity.
[General typographical issues] There are several typographical errors: 'is ometric' in the introduction, the displayed equation at line (24) appears to have a missing superscript on A1, and the repeated expression '(z0 - z0)' in the proof of Theorem 4 suggests widespread missing overlines. A careful proofread of the LaTeX is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the apparent self-application of Theorem 4 in its own proof is a misprint for Theorem 3, and the derivation otherwise rests on independent representation theorems, not on its own conclusion.

full rationale

The paper's main result (Theorem 4) is proved by combining the author's earlier two-operator extension theorem [18, Theorem 1] with a newly proved multidimensional Godič–Lucenko factorization theorem (Theorem 3). Theorem 3 is proved from the model representation Theorem 2 cited from [19]; Theorem 2's assumptions (a commuting SU-set with spectrum of finite multiplicity) do not include the target extension statement, so this self-citation is independent evidence rather than a circular premise. The only literal self-reference in the proof of Theorem 4 occurs twice: 'we may apply Theorem 4 to the operators B_{k,2}' and 'we apply Theorem 4 to the operators U_{j,2}, ... B_{k,2}'. Read literally, this would be circular, but the desired conclusions (22)–(23) are exactly the factorization statement of Theorem 3, and Theorem 4 itself—about symmetric/isometric tuples—cannot be applied to unitary operators B_{k,2}. The context, including the immediately preceding application of Theorem 3 in the τ=0 case and the formulas obtained, identifies these as page-level misprints for Theorem 3. Once corrected, the proof's steps use only hypotheses (a)–(d); the conjugation J is constructed directly in the moment problem section, and no prediction or conclusion is introduced into the assumptions. No load-bearing circularity remains.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; the proof is deterministic. The main external inputs are the author's own published model theorem [19], the two-operator extension theorem [18], and the standard spectral theorem. The theorem hypotheses (a)-(d) and the moment conditions (30) and (B) are domain assumptions of the statements, not derivations.

assumptions (5)

standard math Spectral theorem and joint spectral calculus for commuting families of self-adjoint and unitary operators.
Used throughout, for example in Theorem 2 and in the final spectral measure representation of Theorem 5.
domain assumption The model representation theorem of [19, Theorem 2] for commuting SU-sets with spectrum of finite multiplicity.
Invoked in the proof of Theorem 3 to convert the commuting unitaries on each cyclic subspace into multiplication operators e^{i phi_k} on L2(M_l). This is a published prior result by the same author and is not re-derived here.
standard math Classical Godic-Lucenko theorem: a single unitary operator is the product of two conjugations.
Underlies the rho=2, tau=0 case in [18] and motivates the multidimensional factorization in Theorem 3.
domain assumption Technical hypotheses (a)-(d) of Theorem 4, including essential self-adjointness of A2..A_rho, essential unitarity of B1..B_tau, domain invariance (b), restricted essential self-adjointness (c), and existence of conjugation J in (d).
These are the assumptions of the main extension theorem. Condition (c) makes the Cayley transforms U_{j,2} unitary on H2, and condition (d) is used to build U_{2,4}.
domain assumption For Theorem 5, the moment sequence is assumed to satisfy positivity (30) and, in the multidimensional case, the boundedness condition (B).
These are the hypotheses of the moment problem theorem; they are checked in the application, not derived.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On extensions of commuting tuples of symmetric and isometric operators." pith.science (2026). https://pith.science/paper/KSXJLF3E

@misc{pith2026190800794,
  author       = {Pith},
  title        = {Pith review of: On extensions of commuting tuples of symmetric and isometric operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KSXJLF3E}},
  note         = {Machine review of arXiv:1908.00794}
}

read the original abstract

In this paper we study extensions of commuting tuples of symmetric and isometric operators to commuting tuples of self-adjoint and unitary operators. Some conditions which ensure the existence of such extensions are presented. A multidimensional analog of the Godi\v{c}-Lucenko Theorem is proved. An application to a multidimensional power-trigonometric moment problem is given.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[18]

S. M. Zagorodnyuk, On commuting symmetric operators , Methods Funct. Anal. Topology 18 (2012), no. 2, 198–200

work page 2012
[19]

S. M. Zagorodnyuk, On the density of polynomials in some spaces L2(M ), Mathematical Notes 95 (2014), no. 1, 53–66

work page 2014
[1]

Albrecht, F.-H

E. Albrecht, F.-H. Vasilescu, Unbounded extensions and operator mo- ment problems, J. Funct. Anal. 260 (2011), 2497–2517

work page 2011
[2]

Ju. M. Berezanskii, Expansions in Eigenfunctions of Selfadjoint Op- erators, Amer. Math. Soc., Providence, RI, 1968. (Russian edition: Naukova Dumka, Kiev, 1965)

work page 1968
[3]

Yu. E. Bohonov, On self-adjoint extensions of commuting Hermitian operators, Ukr. matem. zhurnal 42 (1990), no. 5, 695–697. (in Russian)

work page 1990
[4]

E. A. Coddington, Extension theory of formally normal and symmetric subspaces, Memoirs of the American Mathematical Society 134 (1973), 80 pp

work page 1973
[5]

Fuglede, The multidimensional moment problem , Expo

B. Fuglede, The multidimensional moment problem , Expo. Math. 1 (1983), no. 4, 47–65

work page 1983
[6]

S. R. Garcia, M. Putinar, Complex symmetric operators and applica- tions II , Trans. of the Amer. Math. Soc. 359 (2007), 3913–3931

work page 2007

Show all 20 references

[7]

I. M. Gelfand, N. Ya. Vilenkin, Some Applications of Harmonic Anal- ysis. Equipped Hilbert Spaces , Gos. izdat. ﬁz.-mat. lit., Moscow, 1961. (in Russian)

1961
[8]

V. I. Godiˇ c, I. E. Lucenko, On the representation of a unitary operator as a product of two involutions , Uspehi Mat. Nauk 20 (1965), 64–65. (in Russian)

1965
[9]

R. S. Ismagilov, Selfadjoint extensions of commuting symmetric oper- ators, Uspehi matem. nauk XVII (1962), issue 1(103), 177–181. (in Russian). 14

1962
[10]

P. E. T. Jørgensen, Selfadjoint extension operators commuting with an algebra, Math. Z. 169 (1979), 41–62

1979
[11]

A. N. Kochubei, On symmetric operators commuting with a family of unitary operators , Funk. analiz i ego pril. 13 (1979), 4, 77–78. (in Russian)

1979
[12]

R. S. Phillips, The extension of dual subspaces invariant under an algebra, ”Proc. Internat. Sympos. Linear spaces, Jerusalem, 1960” , Jerusalem, Acad Press; Oxford-London-New York-Paris, Per gamon Press, (1961). 366–398. (Russian translation: Matematika 8:6 (1964), 81–108)

1961
[13]

Schm¨ udgen, On commuting unbounded self-adjoint operators, IV , Math

K. Schm¨ udgen, On commuting unbounded self-adjoint operators, IV , Math. Nachr. 125 (1986), 83–102

1986
[14]

S. P. Slinker, On commuting self-adjoint extensions of unbounded oper- ators, Indiana Univ. Math. J. 27 (1978), no. 4, 629–636

1978
[15]

Vasilescu, Operator moment problems in unbounded sets , Oper

F.-H. Vasilescu, Operator moment problems in unbounded sets , Oper. Theory Adv. Appl. 127 (2001), 613–638

2001
[16]

Vasilescu, Quaternionic Cayley transform revisited , J

F.-H. Vasilescu, Quaternionic Cayley transform revisited , J. Math. Anal. Appl. 409 (2014), 790–807

2014
[17]

S. M. Zagorodnyuk, Devinatz’s moment problem: a description of all solutions, Journal of Operator Theory 68 (2012), issue 2, 515–541

2012
[20]

S. M. Zagorodnyuk, Unitary extensions of pairs of commuting isomet- ric operators and their generalized resolvents , New York J. Math. 23 (2017), 555–582. Extensions of commuting tuples of symmetric and isometric operators. S.M. Zagorodnyuk 15 In this paper we study extensions...

2017

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [18]

S. M. Zagorodnyuk, On commuting symmetric operators , Methods Funct. Anal. Topology 18 (2012), no. 2, 198–200

work page 2012

[2] [19]

S. M. Zagorodnyuk, On the density of polynomials in some spaces L2(M ), Mathematical Notes 95 (2014), no. 1, 53–66

work page 2014

[3] [1]

Albrecht, F.-H

E. Albrecht, F.-H. Vasilescu, Unbounded extensions and operator mo- ment problems, J. Funct. Anal. 260 (2011), 2497–2517

work page 2011

[4] [2]

Ju. M. Berezanskii, Expansions in Eigenfunctions of Selfadjoint Op- erators, Amer. Math. Soc., Providence, RI, 1968. (Russian edition: Naukova Dumka, Kiev, 1965)

work page 1968

[5] [3]

Yu. E. Bohonov, On self-adjoint extensions of commuting Hermitian operators, Ukr. matem. zhurnal 42 (1990), no. 5, 695–697. (in Russian)

work page 1990

[6] [4]

E. A. Coddington, Extension theory of formally normal and symmetric subspaces, Memoirs of the American Mathematical Society 134 (1973), 80 pp

work page 1973

[7] [5]

Fuglede, The multidimensional moment problem , Expo

B. Fuglede, The multidimensional moment problem , Expo. Math. 1 (1983), no. 4, 47–65

work page 1983

[8] [6]

S. R. Garcia, M. Putinar, Complex symmetric operators and applica- tions II , Trans. of the Amer. Math. Soc. 359 (2007), 3913–3931

work page 2007