Inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators

Josu\'e I. Rios-Cangas

REVIEW 4 major objections 5 minor 5 references

Inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Definite inner products on Hilbert–Schmidt operators are classified as signed trace series, and the negative term cannot always be removed.

desk verdict Plausible extension of the finite-dimensional classification to Hilbert–Schmidt operators, but the current manuscript has a false theorem and a load-bearing convergence gap in the main proof. read the letter →

arxiv 2506.04541 v1 pith:SDOY4JUB submitted 2025-06-05 math.FA math-phmath.MP

Josué I. Rios-Cangas This is my paper

classification math.FAmath-phmath.MP MSC 46C5046C0547A65

keywords innerproductsHilbert–Schmidtoperatorssesquilinearformspositivenon-negativetracerepresentationselfadjoint

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 proves a classification of the inner products that can be defined on the Hilbert space $S_2$ of Hilbert–Schmidt operators, the bounded operators whose matrix entries are square-summable. The guiding identity is that every continuous sesquilinear form on $S_2$ has the form $\varphi(\eta,\tau)=\operatorname{tr}(\eta^*A\tau)$ for a unique bounded operator $A$, so inner products correspond exactly to positive operators $A$. The main theorem represents every definite inner product as one negative trace term plus a countable series of positive trace terms, and gives an extra condition under which the negative term can be removed and the representation made entirely positive. That negative term is genuinely needed in general: a two-dimensional example shows that no finite all-positive representation exists. This is relevant because $S_2$ is the natural state space for noncommutative quantum mechanics, and knowing all its inner products fixes all admissible quadratic forms on operators.

What carries the argument

The workhorse is the identification of continuous sesquilinear forms on $S_2$ with bounded operators on $S_2$ through $\varphi(\eta,\tau)=\operatorname{tr}(\eta^*A\tau)$, so inner products are exactly positive operators $A$. To analyze positivity, the paper decomposes $A$ using the rank-one matrix units $\varepsilon_{nm}=|e_n\rangle\langle e_m|$, the infinite-dimensional analogues of elementary matrices: Lemma 3.4 writes every $A\in B(S_2)$ as $A\eta=\sum_{n,m}\varepsilon_{nm}\eta a_{nm}$ with $a_{nm}\in B(H)$, and Theorem 3.9 rewrites selfadjoint $A$ as $A\eta=\sum_n a_n\eta b_n$ with selfadjoint $a_n,b_n$. Positivity is then extracted by testing on rank-one operators $|g\rangle\langle f|$, which reduces $\langle\eta,A\eta\rangle_2$ to products $\langle g,a_n g\rangle\langle f,b_n f\rangle$. The one-summand and two-summand cases are settled in Theorems 3.11 and 3.12. Theorem 3.16 then converts the decomposition into the signed representation of a positive definite $A$ by choosing coefficients $\beta_{nm}$ and an auxiliary operator $b_{22}=\varepsilon_{22}-\sum_{(n,m)\ne(1,1),(2,2)}\beta_{nm}\hat{\varepsilon}_{nm}$ that absorb the unwanted cross-terms.

What would settle it

Take a positive definite $A\in B(S_2)$ whose matrix-unit coefficients decay slowly, compute the coefficients $\beta_{nm}$ prescribed by the proof of Theorem 3.16, and check whether the series for $b_{22}=\varepsilon_{22}-\sum_{(n,m)\ne(1,1),(2,2)}\beta_{nm}\hat{\varepsilon}_{nm}$ converges in operator norm. If a legitimate positive definite $A$ makes the series diverge, the proof of the representation theorem does not cover that case; if the series can be shown always to converge, the gap is closed.

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Extended reading notes

Core claim

The central claim is Theorem 3.21: if $\varphi:S_2\times S_2\to\mathbb{C}$ is a definite inner product, then there exist non-negative operators $a_3,\dots$ and positive definite operators $a_1,a_2,b_1,\dots$ in $B(H)$ such that $\varphi(\eta,\tau)=-\operatorname{tr}(\eta^*a_1\tau b_1)+\sum_{n\ge 2}\operatorname{tr}(\eta^*a_n\tau b_n)$ for all $\eta,\tau\in S_2$. If, in addition, there are $\zeta_n>0$ with $b_n-\zeta_n b_1$ positive definite and $-a_1+\sum_{n\ge2}\zeta_n a_n$ non-negative, then the same inner product has an all-positive representation $\varphi(\eta,\tau)=\sum_n\operatorname{tr}(\eta^*a_n\tau b_n)$ with $\cap_n\ker a_n=\{0\}$ and all $b_n$ positive definite. The paper also proves the converse sufficiency direction in Theorem 3.19: any series of this all-positive form with trivial joint kernel and positive $b_n$'s is an inner product. The negative sign is not removable in general; Counterexample 3.13 exhibits a positive definite inner product on a two-dimensional space that cannot be written as a finite sum of all-positive trace terms.

Load-bearing premise

The main theorem's proof depends on certain infinite sums of simple finite-rank operators, for example the auxiliary operator $b_{22}$, actually defining bounded operators on the underlying Hilbert space, and the paper does not prove this convergence.

Editorial extensions

If this is right

Every definite inner product on $S_2$ has a concrete normal form: one negative trace term plus a countable positive trace series, with all operators in $B(H)$.
If the $\zeta$-condition holds, the same inner product can be rewritten entirely with positive terms, so the minus sign is a feature of the chosen decomposition rather than of the inner product itself.
Counterexample 3.13 shows that no finite all-positive representation exists in general, so the signed form is not merely a technical artifact.
The sufficiency direction gives a recipe for constructing inner products on $S_2$: choose non-negative $a_n$ with $\cap_n\ker a_n=\{0\}$ and positive $b_n$, and the trace series defines an inner product.
For the finite cases $m=1,2$, definite inner products always have all-positive representations (Theorem 3.20).

Reading between the lines

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

One consequence left implicit is that the signed representation functions as a normal form for positive operators on $S_2$: the coefficient decomposition gives a positivity certificate, and the $\zeta$-condition could be used as a practical test for whether an operator admits an all-positive trace decomposition.
The paper establishes only sufficiency of the $\zeta$-condition, so whether the condition is also necessary for an all-positive representation remains open; the boundary between the signed and unsigned cases is not yet characterized.
A testable extension is to allow infinitely many positive summands for the two-dimensional counterexample; the example rules out finite sums only, so an infinite all-positive representation of that inner product is not excluded by the paper's arguments.

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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

Here's my read. The paper extends [4]'s classification of inner products on matrix spaces to the infinite-dimensional Hilbert–Schmidt setting. The main result, Theorem 3.21, says any definite continuous inner product on S2 is a sum of traces with at most one negative term, and gives a condition for an all-positive representation. If true, that's a useful structural theorem for non-commutative quantum mechanics and operator theory.

What's genuinely good: the representation theorems for operators on B(S2) (Theorems 3.6 and 3.9) are sensible. The sufficiency direction (Theorem 3.19) is correct under its hypotheses, and Counterexample 3.13 correctly shows the obstruction to an all-positive representation. The paper is clearly organized and the debt to [4] is acknowledged honestly.

The soft spots are real. Theorem 3.2(iv) is false as stated. Take H=l2, a_n=I, b_n=lambda_n |e_n>0 and lambda_n->0. Then m_a=1>0 and the intersection of the kernels of the b_n is {0}, but A eta = eta diag(lambda_n) is only non-negative, not positive definite. The proof's termwise lower bound does not control the tail.

Corollary 2.3 is overbroad: it claims all inner-product norms on a Banach space are equivalent, but the proof relies on Lemma 2.2, which requires a Hilbert space and continuity. The displayed inequality in the proof also has a dimension error (a squared norm where a norm belongs).

The serious issue is in Theorem 3.16. The operator b22 = epsilon22 - sum_{nm != (1,1),(2,2)} beta_nm hat(epsilon_nm) is introduced as an infinite sum with coefficients beta_nm chosen 'sufficiently large.' No convergence in B(H) is shown. Since each hat(epsilon_nm) has operator norm at least 1/sqrt(2) for n != m, the coefficients must decay for the series to define a bounded operator; the proof gives no decay condition. The subsequent algebra and the final classification in Theorem 3.21 depend on this b22, so the central proof is incomplete.

My overall verdict: the strategy is plausible and the finite-dimensional base makes the result believable, but the current manuscript is not rigorous. The false Theorem 3.2(iv) and the missing convergence proof are repairable in principle, but they need real work. I would send this to a serious referee and ask for major revision. I would not cite it in its current form.

Referee Report

4 major / 5 minor

Summary. The paper aims to characterize all continuous sesquilinear forms and inner products on the Hilbert space S2 of Hilbert–Schmidt operators. It first reviews the representation of continuous sesquilinear forms on a Hilbert space by bounded operators, then develops decompositions of operators in B(S2) as sums η ↦ Σ a_n η b_n, and gives positivity criteria for such operators. The central result, Theorem 3.21, asserts that every definite inner product on S2 has the form φ(η,τ) = -tr(η* a1 τ b1) + Σ_{n≥2} tr(η* a_n τ b_n), with an all-positive representation under an additional domination condition. The paper also contains a finite-dimensional counterexample, taken from [4], showing that a negative term is sometimes unavoidable.

Significance. If the main classification were established, it would be a useful infinite-dimensional extension of the authors' earlier finite-dimensional work and would clarify the possible forms of inner products on the Hilbert–Schmidt space, with potential relevance to noncommutative quantum mechanics. The paper has a clear organizational strategy, and the identification of inner products with positive operators is a natural framework. However, the central proof depends on infinite sums of rank-one operators for which no convergence in B(H) is proved, and two supporting results are false as stated. The counterexample in Counterexample 3.13 is a useful concrete input, but the main positive claims are not yet supported by rigorous arguments.

major comments (4)

[§2, Corollary 2.3] The statement is false as written, and the proof contains dimensionally inconsistent inequalities. On H = ℓ2, the operator T = diag(1/n) is positive in the sense of (2.1) and belongs to B(H), so φ(f,g) = ⟨Tf,g⟩ is an inner product in the paper's own terminology. Its norm is not equivalent to the usual norm because ||e_n||_φ = 1/√n while ||e_n|| = 1. The proof writes ||f||_{φ1}^2 ≤ ||f||_{φ2}^2 ||T1|| and then ∥f∥_{φ2} ≤ ∥f∥^2_{φ1} ∥T2∥, mixing first and second powers; the chain cannot yield the claimed equivalence. Lemma 2.2 and Corollary 2.3 must be restricted to continuous inner products that are equivalent to the given norm, or reformulated using positive definite operators with bounded inverse.
[§3.1, Theorem 3.2(iv)] Theorem 3.2(iv) is false as stated. The proof shows ⟨η,Aη⟩ ≥ ∥a_j^{1/2} η b_j^{1/2}∥_2^2 ≥ m_{a_j} ∥η b_j^{1/2}∥_2^2, but ∥η b_j^{1/2}∥_2 can be arbitrarily small even when ∥η∥_2 = 1. Concretely, take H = ℓ2, a_n = I, and b_n = λ_n |e_n⟩⟨e_n| with λ_n > 0 and λ_n → 0. Then m_{a_n} = 1, m_a = 1, and ∩_n ker b_n = {0}, so the hypotheses hold, yet Aη = η diag(λ_n) is positive but not positive definite. This invalidates the criterion as stated; either the hypothesis needs a uniform lower bound forcing ∥η b_j^{1/2}∥_2 ≥ c∥η∥_2, or the conclusion must be weakened.
[§3.2, Lemma 3.15 and Theorem 3.16] The proof of the central characterization introduces infinite sums of rank-one operators without proving convergence in B(H). In Lemma 3.15, the operator γ11(ε11 + t0 ε22) + Σ_{(n,m)≠(1,1),(2,2)} γ_{nm} ε̂_{nm} is asserted to be positive definite in B(H), but no argument shows that the infinite series converges in operator norm; the coefficients γ_{nm} are only known to be bounded, not summable or square-summable. In Theorem 3.16, the operator b22 = ε22 − Σ_{(n,m)≠(1,1),(2,2)} β_{nm} ε̂_{nm} is treated as an element of B(H), with β_{nm} chosen 'sufficiently large' to make β_{nm} a22 + a_{nm} positive definite. Since ∥ε̂_{nm}∥ ≥ 1/√2 for n ≠ m and no decay of β_{nm} is established, the series may diverge in norm, and Aη cannot legitimately be rewritten as in (3.31). The boundedness of A ∈ B(S2) implies Hilbert–Schmidt constraints on the blocks a_{nm}, but it does not force the stronger matrix boundedness of the admissible β_{nm}. This is a load-bearing gap: (3.31), and therefore (3.28) and Theorem 3.21, are not established.
[§3.3, Theorem 3.21] Theorem 3.21, the main classification, is proved by a direct appeal to Lemma 3.1 and Theorem 3.16. Since Theorem 3.16 is not established for the reasons above, the final classification is unsupported. In addition, the infinite series in (3.28) and (3.30) is introduced without a convergence argument in B(S2); it is not enough to know that each term tr(η* a_n τ b_n) is finite. A revised proof must either prove convergence of all infinite sums involved or state and prove an appropriate summability condition coming from A ∈ B(S2).

minor comments (5)

[§2] The wording 'all inner products' in Lemma 2.2 should be qualified: only inner products that are continuous with respect to the given Hilbert space norm are represented by operators in B(H). Otherwise the statement is false, as shown by the example T = diag(1/n) in relation to Corollary 2.3.
[§3.2, Theorem 3.9] The derivation of (3.14) is hard to follow: the displayed formula contains redundant cancellations and no explanation of how the double series is rearranged into a single countable sum. Please expand this step.
[§3.2, Theorem 3.20(ii)] The conclusion says 'whit kera1∩kera2 = {0}', but the operators in the conclusion are renamed as â1, â2 and new b1, b2; the kernel condition should be stated for the renamed operators, and the typo 'whit' should be corrected.
[Throughout] There are several typographical errors: 'stablished' in the paragraph after Remark 2.1, 'well-know' at the beginning of Section 3, 'Lemm 3.15' in the proof of Theorem 3.16, and garbled 'CauchyâĂŞSchwarz' in two places. The paragraph after Theorem 3.2 uses 'j ̸=k' where h ≠ k is meant.
[§3.2, Theorem 3.16] The notation in the proof of Theorem 3.16 changes meaning several times: a_{nm} is first a set of selfadjoint operators from Lemma 3.15, then after (3.31) it denotes different operators including positive definite ones. The renaming steps should be made explicit to allow the reader to verify the algebra.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the only self-citation is the motivational finite-dimensional counterexample from [4], which is not used in the proof of the central theorem.

full rationale

The paper's derivation is a self-contained functional-analytic argument. Lemma 2.2 and Lemma 3.1 are standard correspondences; Theorems 3.6 and 3.9 give decompositions via the canonical basis; the positivity criteria in Theorems 3.11, 3.12, 3.15, and 3.16 are proved in the paper by rearrangement and polarization-type identities. The central representation, Theorem 3.21, follows formally from Theorem 3.16, and the 'if' part is an algebraic reparameterization in (3.32), not a fitted or assumed conclusion. The finite-dimensional Counterexample 3.13 is cited from the authors' earlier paper [4] and motivates the appearance of the negative term, but it is not invoked in the proofs of Theorems 3.16 or 3.21. The construction of b22 as an infinite sum in the proof of Theorem 3.16 has a convergence and boundedness gap; that is a correctness risk, not a circularity, because the claimed equality is still intended to be derived from the given positive-definite A rather than assumed. No step reduces to its inputs by definition, and no fitted quantity is renamed as a prediction. The only self-citation is minor and motivational, not load-bearing.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard functional analysis and on the specific infinite-sum representations constructed in the proof. There are no empirical free parameters and no new postulated physical entities. The main unstated load-bearing assumptions are the convergence and interchange-of-limits properties of the countably infinite sums of finite-rank operators.

assumptions (4)

domain assumption All inner products and sesquilinear forms under consideration are continuous with respect to the original Hilbert-space norm, and the Hilbert space remains complete under each new inner-product norm when needed.
Lemma 2.2 and Corollary 2.3 rely on the Riesz representation theorem for continuous forms. Without continuity or completeness, strictly positive operators with spectrum accumulating at zero give inner products whose norms are not equivalent to the original norm.
standard math The rank-one operators εnm form an orthonormal basis of S2, and every A∈B(S2) admits an expansion Aη=Σ εnm η anm.
This is the foundation of Section 3 and is used in Lemma 3.4 and Theorem 3.6.
ad hoc to paper The series of finite-rank operators introduced in Lemma 3.15 and Theorem 3.16 converge to bounded operators on H, for example b22=ε22−Σ_{(n,m)≠(1,1),(2,2)} βnm \u005cu005cu005cvarepsilon_nm.
This convergence is asserted without proof; boundedness of A∈B(S2) does not automatically imply that the scalar matrix (βnm) defines a bounded operator on H.
domain assumption Passing limits through the infinite sums in Lemma 3.15 is justified, including the use of dominated convergence on the quadratic forms.
The paper uses limits of sequences ⟨fj, anm fj⟩ to define operators γnm \u005cu005cu005cvarepsilon_nm and then treats the resulting infinite combinations as bounded positive operators.

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Pith. "Pith review of Inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators." pith.science (2026). https://pith.science/paper/SDOY4JUB

@misc{pith2026250604541,
  author       = {Pith},
  title        = {Pith review of: Inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SDOY4JUB}},
  note         = {Machine review of arXiv:2506.04541}
}

abstract

This work presents a rigorous characterization of inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators. We first deal with a general setting of continuous sesquilinear forms on a Hilbert space $\mathcal H$, and provide a characterization of all inner products by means of positive operators in $\mathcal {B(H)}$. Next, we establish necessary and sufficient conditions for an operator in $\mathcal B(S_2)$ to be positive. Identifying an inner product with a positive operator enables us to rigorously describe inner products on $S_2$.

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Works this paper leans on

5 extracted references · 5 canonical work pages

[4]

Mart´ ınez-Avenda˜ no and Josu´ e I

Rub´ en A. Mart´ ınez-Avenda˜ no and Josu´ e I. Rios-Cangas,Inner products on the space of complex square matrices, Linear Algebra Appl. 439 (2013), no. 11, 3620–3637. MR 3119877

work page 2013
[1]

Isiaka Aremua, Ezinvi Balo¨ ıtcha, Mahouton Norbert Hounkonnou, and Komi Sodoga,On Hilbert-Schmidt operator formulation of noncommutative quantum mechanics , Mathematical structures and applications, STEAM-H: Sci. Technol. Eng. Agric. Math. Health, Springer, Cham, 2018, pp. 61–118. MR 3840084

work page 2018
[2]

M. Sh. Birman and M. Z. Solomjak, Spectral theory of selfadjoint operators in Hilbert space , Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, 1987, Translated from the 1980 Russian original by S. Khrushch¨ ev and V. Peller. MR 1192782 (93g:47001)

work page 1987
[3]

Dhruba Prakash Biswas, Priti Sharma, and Sandip Jana, Characterisation of equivalent norms on a linear space using exponential vector space , South East Asian J. Math. Math. Sci. 19 (2023), no. 3, 87–100. MR 4726393

work page 2023
[5]

265, Springer, Dordrecht, 2012

Konrad Schm¨ udgen,Unbounded self-adjoint operators on Hilbert space , Graduate Texts in Mathematics, vol. 265, Springer, Dordrecht, 2012. MR 2953553 Departamento de Matem´aticas, Universidad Aut´onoma Metropolitana, Unidad Iztapalapa, San Rafael Atlixco 186, 093110 Iztapalapa, Ciudad de M´exico. Email address: jottsmok@xanum.uam.mx 14

work page 2012

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [4]

Mart´ ınez-Avenda˜ no and Josu´ e I

Rub´ en A. Mart´ ınez-Avenda˜ no and Josu´ e I. Rios-Cangas,Inner products on the space of complex square matrices, Linear Algebra Appl. 439 (2013), no. 11, 3620–3637. MR 3119877

work page 2013

[2] [1]

Isiaka Aremua, Ezinvi Balo¨ ıtcha, Mahouton Norbert Hounkonnou, and Komi Sodoga,On Hilbert-Schmidt operator formulation of noncommutative quantum mechanics , Mathematical structures and applications, STEAM-H: Sci. Technol. Eng. Agric. Math. Health, Springer, Cham, 2018, pp. 61–118. MR 3840084

work page 2018

[3] [2]

M. Sh. Birman and M. Z. Solomjak, Spectral theory of selfadjoint operators in Hilbert space , Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, 1987, Translated from the 1980 Russian original by S. Khrushch¨ ev and V. Peller. MR 1192782 (93g:47001)

work page 1987

[4] [3]

Dhruba Prakash Biswas, Priti Sharma, and Sandip Jana, Characterisation of equivalent norms on a linear space using exponential vector space , South East Asian J. Math. Math. Sci. 19 (2023), no. 3, 87–100. MR 4726393

work page 2023

[5] [5]

265, Springer, Dordrecht, 2012

Konrad Schm¨ udgen,Unbounded self-adjoint operators on Hilbert space , Graduate Texts in Mathematics, vol. 265, Springer, Dordrecht, 2012. MR 2953553 Departamento de Matem´aticas, Universidad Aut´onoma Metropolitana, Unidad Iztapalapa, San Rafael Atlixco 186, 093110 Iztapalapa, Ciudad de M´exico. Email address: jottsmok@xanum.uam.mx 14

work page 2012