{"id":"f051d2da-4a13-4130-8891-ea50c464549c","arxiv_id":"2608.13550","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Positive Toeplitz operators on pluriharmonic Fock space belong to the same Schatten classes as their holomorphic counterparts, with sharp norm comparison constants and an exact trace identity.","lead":"The paper proves that positive Toeplitz operators on pluriharmonic Fock space are in the Schatten class S_p exactly when the corresponding holomorphic Toeplitz operator is, for every 0<p<infinity, settling a conjecture in several complex variables. It also gives the sharp constant (a factor 2 to the max(1,p)) comparing the two Schatten norms, and an exact trace formula, which makes the result useful for anyone working with Fock-space operators.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.1's pointwise kernel formula (14) is missing a complex conjugation; as printed it makes point-mass Toeplitz operators anti-holomorphic, so the bridge from the form convention to the holomorphic criterion is false as stated.","rationale":"The paper's central contribution is a clean reduction of pluriharmonic Schatten membership to the holomorphic case via a two-block decomposition. I checked the two load-bearing steps: the imported holomorphic criterion is used honestly, since the normalization and the auxiliary integrability implication (16) are verified, and the positive-block principle is sound, because B_j=B^{1/2}P_j B^{1/2} has the same nonzero singular values as A_j via Y_jY_j^*=iota_j A_j iota_j^*. The Schatten comparison and the optimality examples are consistent with the stated theorem. The problem is in the bridge lemma, not in the main argument: Lemma 3.1 states the wrong pointwise kernel formula. For a point mass at a neq 0, the printed formula returns an anti-holomorphic function, contradicting both the form operator and the point-mass calculation in Proposition 6.2. This is an internal inconsistency in the proof text rather than a disagreement with consensus. The fix is local: replace K^H_z(w) by its conjugate in (14) and in the corresponding line of the converse proof. With that correction the proof is complete. I therefore recommend conditional acceptance rather than rejection: the theorem should survive, but the text as printed needs the correction before publication. The reader's weakest assumption (the imported holomorphic criterion) is reasonable and I do not dispute it; the more immediate blocker is the conjugation error in Lemma 3.1.","tokens_in":13191,"tokens_out":23715,"duration_ms":219479,"concrete_test":"Take H=F^2_alpha, a neq 0, mu=delta_a, and f=1. Compute the form operator S via (3): S f(z)=e^{-|a|^2/alpha} K_a(z)=e^{z dot bar a / alpha - |a|^2/alpha}, which is holomorphic. Compare with Lemma 3.1's formula (14) as printed: T_{mu,0} f(z)=e^{-|a|^2/alpha} f(a) K_z(a)=e^{a dot bar z / alpha - |a|^2/alpha}, which is anti-holomorphic and not in F^2_alpha. Re-run the same test with overline{K^H_z(w)} in place of K^H_z(w); the formula then reproduces the holomorphic expression S f(z), confirming the missing conjugation.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Lemma 3.1 is printed with the wrong kernel in (14). For H=F^2_alpha the reproducing kernel is K^H_z(w)=e^{w dot bar z / alpha}; the form operator for mu=delta_a, a neq 0, satisfies (S f)(z)=e^{-|a|^2/alpha} f(a) K_a(z)=e^{-|a|^2/alpha} f(a) e^{z dot bar a / alpha}, a holomorphic function of z. The displayed formula (14) gives the integral of f(w) K^H_z(w), which for this measure equals e^{-|a|^2/alpha} f(a) K_z(a)=e^{-|a|^2/alpha} f(a) e^{a dot bar z / alpha}, an anti-holomorphic function of z that is not in F^2_alpha. The converse implication in Lemma 3.1 is therefore false as stated; the missing step is the complex conjugate overline{K^H_z(w)} in the integrand. Lemma 3.2's identification of T^0 with c T relies on this kernel bridge, so as printed the proof has a false lemma at a load-bearing point. The intended statement is evident from Proposition 6.2's point-mass calculation, which uses K_a(z), not K_z(a); with the conjugate inserted, the proof of Theorem 1.1 is sound.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies positive Toeplitz operators on the pluriharmonic Fock space PH^2_α over C^n. It proves that, for every 0<p<∞ and any r>0, the pluriharmonic Toeplitz operator T_μ^ph belongs to the Schatten class S_p if and only if the holomorphic Toeplitz operator T_μ does, and if and only if the ball-average function z↦μ(B(z,r)) lies in L^p(C^n,dV). This settles a conjecture of Jaguzović and Vujadinović for n≥2 and covers the full range 0<p<1 in every dimension. The paper also establishes the sharp norm comparison 1≤(‖T_μ^ph‖_{S_p}/‖T_μ‖_{S_p})^p≤2^{max(1,p)} with optimal constants, extends the comparison to all symmetrically normed ideals, and derives an exact trace identity for the trace-class case. The main technical ingredients are an orthogonal splitting of PH^2_α into holomorphic and anti-holomorphic parts, a general positive-block principle for operators on orthogonal sums, and the holomorphic Schatten criterion imported from Isralowitz–Virtanen–Wolf.","tokens_in":13494,"tokens_out":13128,"duration_ms":112157,"significance":"If the technical issues noted below are corrected, this is a substantial contribution. The main theorem is a complete answer to the pluriharmonic analogue of the Fock-space Schatten criterion, including the previously open range 0<p<1 and the sharp constants. The positive-block principle (Proposition 4.1) is a clean and general tool that is likely to be useful beyond the present setting, and the exact trace identity gives a quantitative refinement not present in earlier work. The paper is well written overall, with most central claims proven in detail and with explicit references for the external results it uses.","major_comments":[{"comment":"The displayed kernel formula in (14) is missing the complex conjugation: the integrand should be f(w)\\overline{K_z^H(w)}e^{-|w|^2/\\alpha}\\,d\\mu(w), not f(w)K_z^H(w)e^{-|w|^2/\\alpha}\\,d\\mu(w). As printed, for H=F^2_\\alpha and μ=δ_a, the operator maps f to f(a)e^{-|a|^2/\\alpha}K_z(a), which is anti-holomorphic in z and does not lie in F^2_\\alpha; the lemma is therefore false as stated. The proof itself uses the conjugated kernel in the converse direction (it writes (Sf)(z)=⟨Sf,K_z⟩=∫ f(w)\\overline{K_z(w)}dμ(w)), so the error is a missing conjugate in the display. This is load-bearing because Lemma 3.2 relies on this bridge to identify the form-defined operator T_μ with the kernel-defined operator of [8]; with the conjugate inserted, the intended statement is correct and the proof goes through.","section":"§3, Lemma 3.1, Eq. (14)"},{"comment":"The displayed computation of the second diagonal block contains incorrect equalities. The form restricted to the second summand should be q(ι_2h,ι_2k)=∫(Jh)(z)\\overline{(Jk)(z)}e^{-|z|^2/\\alpha}d\\mu(z)=∫\\overline{h(z)}k(z)e^{-|z|^2/\\alpha}d\\mu(z); the printed text instead writes this as ∫ h(z)k(z)dμ and then as ⟨Ah,k⟩, which is not equal to the preceding expression in general. The final conclusion D=JA_0J^{-1} is correct (for instance, one can verify ⟨D\\bar h,\\bar k⟩=⟨A_0k,h⟩ for h,k∈F^2_{α,0}), but the proof as written is invalid. Since Lemma 5.1 supplies the singular value bound (30) used in the proof of Theorem 1.1, this computation must be rewritten.","section":"§5, Lemma 5.1, proof"},{"comment":"In the statement of Lemma 5.1, the map J is described as J:F^2_{α,0}→F^2_{α,0}, Jh=\\bar h. Since \\bar h is anti-holomorphic, the codomain should be the conjugate space \\overline{F^2_{α,0}} (the second summand in the orthogonal splitting). This is a notational error, but it is related to the proof issue above and should be corrected to avoid confusion.","section":"§5, Lemma 5.1, statement"}],"minor_comments":[{"comment":"The notation \\overline{F^2_{α,0}} for the conjugate space is introduced twice and could be defined explicitly as the set of conjugates of functions in F^2_{α,0} to avoid ambiguity, especially since the same overline symbol also denotes complex conjugation of individual functions.","section":"§2 and Introduction"},{"comment":"In the proof for 0<p<1, the sentence 'For these measures, every nonconstant eigenvalue occurs with twice its holomorphic multiplicity' is terse; the subsequent calculation makes it clear, but a one-sentence explanation of why the mixed blocks vanish would improve readability.","section":"§6, Proposition 6.2"},{"comment":"There are occasional typographical spacing errors in the title and running header (e.g., 'OPERA TORS', 'SP ACE'); these are presumably typesetting artifacts and should be corrected in the final version.","section":"Title and header"},{"comment":"Reference [7] is cited as an arXiv preprint; if a published version is available, it would be helpful to update the citation.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The first reader's report appears overly generous in light of the missing conjugation in Lemma 3.1. The error is genuine and is not merely a typo in a peripheral remark: the lemma's proof is internally inconsistent with the displayed formula, and the lemma is used to connect the form convention to the holomorphic Schatten criterion. However, the intended statement is clear and the fix is local, so I recommend major revision rather than rejection. The block-principle argument and the sharp constant computations are otherwise convincing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Jaguzović–Vujadinović conjecture is settled for n≥2, and the full range 0<p<∞ is covered in every dimension. The sharp norm comparison (1 vs 2^{max(1,p)}) and the trace identity are genuinely new. The positive block principle (Prop. 4.1) is a nice piece of work—it avoids radiality assumptions and cleanly handles the mixed block. The proof of the equivalence is carefully structured: orthogonal splitting (Lemma 2.1), identification of the second diagonal block (Lemma 5.1), and optimality via point masses (p≥1) and circle measures (p<1) all check out. The reliance on the known holomorphic Schatten criterion [8] is clearly disclosed and the auxiliary integrability hypothesis is verified. No circularity; the cited results are prior, independent, and standard.\n\nThe one soft spot is Lemma 3.1. As printed, the kernel in (14) is missing a complex conjugation: it writes f(w) K^H_z(w) dµ(w), which for point masses produces anti-holomorphic functions and makes the statement false. The intended formula is f(w) \\overline{K^H_z(w)}, and the proof's converse direction actually uses that. This is a typo, but it sits at a load-bearing point because Lemma 3.1 is what identifies the form-defined operator with the kernel-defined operator used in Lemma 3.2. The fix is trivial and the rest of the argument is unaffected. Everything downstream, including Proposition 6.2, uses the correct kernel.\n\nThis paper deserves a serious referee. It resolves a recently posed conjecture, extends to 0<p<1, and gives sharp constants. The presentation is a bit rough (OCR artifacts, the conjugate-space notation), but the mathematics is sound once the typo is fixed. I'd want the referee to check the normalization matching in Lemma 3.2 carefully and confirm the trace identity (which I did—it's fine). I'd probably not cite it until the corrected version appears, but the result is reliable.","headline":"Solid resolution of the conjecture with a correctable typo in Lemma 3.1—send to review with a request to fix the kernel formula.","tokens_in":14012,"tokens_out":7475,"would_cite":true,"duration_ms":62826,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47B35","47B10","46E22","31C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a positive Toeplitz operator on pluriharmonic Fock space belongs to the Schatten class $S_p$ exactly when the measure's ball masses lie in $L^p$, for every $0<p<\\infty$.","keywords":["Pluriharmonic Fock space","Toeplitz operators","Schatten class","positive measure","symmetrically normed ideal","Carleson measure","singular values","trace identity"],"falsifier":"The theorem would be false if some positive Borel measure $\\mu$ had $\\mu(B(z,r))\\in L^p(\\mathbb{C}^n,dV)$ for some $r>0$ while $T_\\mu^{\\mathrm{ph}}\\notin S_p$, or if any choice of $\\mu$ and $p$ produced a norm ratio outside $[1,2^{\\max\\{1,p\\}}]$. A direct check would compute the singular values of $T_\\mu^{\\mathrm{ph}}$ for a finite weighted sum of point masses and compare the resulting $S_p$ norm with the ball-mass $L^p$ norm.","tokens_in":13006,"feed_emoji":"🎯","tokens_out":5143,"duration_ms":54345,"temperature":0.7,"pith_summary":"The paper establishes a complete Schatten-class criterion for positive Toeplitz operators on pluriharmonic Fock space over $\\mathbb{C}^n$. It shows that for every $0<p<\\infty$, the operator $T_\\mu^{\\mathrm{ph}}$ belongs to $S_p$ if and only if the holomorphic Toeplitz operator $T_\\mu$ does, and if and only if the function $z\\mapsto\\mu(B(z,r))$ lies in $L^p$ for one, equivalently every, radius $r>0$. This settles a conjecture raised for $n\\geq 2$ and extends the criterion to the full range $0<p<1$ in every dimension. The same proof supplies sharp norm comparisons between the pluriharmonic and holomorphic operators, with optimal constants $1$ and $2^{\\max\\{1,p\\}}$, and extends to every symmetrically normed ideal.","feed_headline":"Ball masses decide Schatten class on pluriharmonic Fock space","feed_subtitle":"A measure's fixed-radius ball masses in L^p characterize S_p membership for all 0<p<∞, with sharp norm bounds.","key_machinery":"The central mechanism is the orthogonal splitting $PH^2_\\alpha = F^2_\\alpha \\oplus \\overline{F^2_{\\alpha,0}}$, under which $T_\\mu^{\\mathrm{ph}}$ has a positive $2\\times 2$ block form whose diagonal blocks are $T_\\mu$ and an antiunitary copy of the compression $P_0 T_\\mu|_{F^2_{\\alpha,0}}$. A positive-block principle, Proposition 4.1, says that for a positive form with diagonal operators $A_j$, the representing operator $B$ equals $\\sum_j B^{1/2}P_jB^{1/2}$, each summand has the same nonzero singular values as the corresponding $A_j$, and submajorization gives $\\sum_{k=1}^N s_k(B)\\leq \\sum_j\\sum_{k=1}^N s_k(A_j)$. This yields the Schatten, ideal, and trace estimates without computing the mixed blocks.","core_discovery":"For a positive Borel measure $\\mu$ on $\\mathbb{C}^n$, the pluriharmonic Toeplitz operator $T_\\mu^{\\mathrm{ph}}$ acting on $PH^2_\\alpha$ is in the Schatten class $S_p$ precisely when the local mass function $\\mu(B(\\cdot,r))$ is in $L^p(\\mathbb{C}^n,dV)$, for any fixed $r>0$. Equivalently, $T_\\mu^{\\mathrm{ph}}\\in S_p$ if and only if the holomorphic Toeplitz operator $T_\\mu\\in S_p$. Whenever these hold, the Schatten $p$-norms satisfy $\\|T_\\mu\\|_{S_p}^p \\leq \\|T_\\mu^{\\mathrm{ph}}\\|_{S_p}^p \\leq 2^{\\max\\{1,p\\}}\\|T_\\mu\\|_{S_p}^p$, and both constants are optimal. The paper also proves an exact trace identity, $\\|T_\\mu^{\\mathrm{ph}}\\|_{S_1} = 2\\mu(\\mathbb{C}^n) - \\int_{\\mathbb{C}^n} e^{-|z|^2/\\alpha}\\,d\\mu(z)$, and a sharp two-sided comparison for all symmetrically normed ideals.","pith_inferences":["The same positive-block principle might apply to other orthogonal splittings of reproducing kernel Hilbert spaces, for example weighted pluriharmonic spaces with non-Gaussian weights, where a local-ball criterion would be a testable extension.","The optimality examples suggest that for $0<p<1$ the extremal ratio $2$ is approached by measures spread over circles, so finite-rank approximations of such measures could give explicit near-extremal configurations.","Because positivity is essential, a natural next problem is a signed-measure analogue: for non-positive symbols the mixed block is no longer controlled by diagonal blocks, so any Schatten criterion would need a genuinely different mechanism.","The trace identity could be used as a numerical check in concrete examples, for instance comparing the eigenvalue sums of $T_\\mu$ and $T_\\mu^{\\mathrm{ph}}$ for point-mass or arc-measure symbols."],"forward_implications":["The Schatten-class characterization on pluriharmonic Fock space is now complete: membership is decided by a uniformly local ball-mass condition, with no additional hypotheses on the measure.","The sharp norm comparison gives a two-sided control of singular values: the pluriharmonic operator is never cheaper than the holomorphic one, and its $p$-th Schatten norm is at most $2^{\\max\\{1,p\\}}$ times larger.","For every symmetrically normed ideal, membership of $T_\\mu^{\\mathrm{ph}}$ is equivalent to membership of $T_\\mu$, with a universal constant $2$ in the ideal norm.","The exact trace identity pins down the trace-class case: the trace of $T_\\mu^{\\mathrm{ph}}$ is twice the total mass of $\\mu$ minus the Gaussian-weighted mass, so the ratio can take every value in $[1,2)$.","The proof removes the radiality restriction that appeared in earlier sufficiency arguments, because the mixed block is controlled by positivity rather than by explicit computation."],"supporting_citations":[{"why":"Posed the pluriharmonic Schatten conjecture and proved the $p=1$ case, necessity for $p>1$, and sufficiency for radial measures.","marker":"[10]"},{"why":"Supplies the holomorphic Schatten criterion in generalized Fock spaces that Lemma 3.2 imports and normalizes.","marker":"[8]"},{"why":"Gives the one-dimensional holomorphic Fock-space criterion and trace formula that the paper extends to pluriharmonic spaces.","marker":"[9]"},{"why":"Provides the finite-dimensional positive block decomposition and symmetric-norm consequences used in Proposition 4.1.","marker":"[1]"},{"why":"Rotfel'd's inequality underlies the trace estimates for positive block operators.","marker":"[13]"},{"why":"Supplies the theory of symmetrically normed ideals and the singular-value ideal inequality used in Section 4.","marker":"[15]"},{"why":"McCarthy's $p$-triangle inequality controls quasi-norms for $0<p<1$ in the block estimate.","marker":"[11]"},{"why":"Ky Fan's inequality gives the submajorization estimate for sums of positive compact operators.","marker":"[5]"},{"why":"Earlier block-matrix decomposition for pluriharmonic function spaces sets the framework used in the splitting.","marker":"[6]"},{"why":"Provides the sesquilinear-form convention for singular Toeplitz symbols that the paper adopts.","marker":"[14]"}],"fun_headline_variants":["Ball mass L^p test settles Schatten class on pluriharmonic Fock","Sharp norm bounds for pluriharmonic Toeplitz in Schatten ideals","Settled conjecture: ball mass in L^p iff Schatten class","Exact trace identity for pluriharmonic Toeplitz operators","All p>0: ball mass in L^p iff Schatten membership"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire equivalence leans on the imported holomorphic Schatten criterion saying a holomorphic Toeplitz operator is in $S_p$ exactly when its ball-mass function is in $L^p$; the paper verifies only the normalization and an auxiliary integrability condition, not that theorem itself.","fun_headline_variants_meta":{"raw":{"variants":["Ball mass L^p test settles Schatten class on pluriharmonic Fock","Sharp norm bounds for pluriharmonic Toeplitz in Schatten ideals","Settled conjecture: ball mass in L^p iff Schatten class","Exact trace identity for pluriharmonic Toeplitz operators","All p>0: ball mass in L^p iff Schatten membership"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001194,"raw_usage":{"total_tokens":4994,"prompt_tokens":1083,"completion_tokens":3911,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":699,"completion_tokens_details":{"reasoning_tokens":3808}},"tokens_in":699,"tokens_out":3911,"duration_ms":31643,"temperature":1.0,"reasoning_tokens":3808,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:19:07.034754+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The theorem would be false if some positive Borel measure $\\mu$ had $\\mu(B(z,r))\\in L^p(\\mathbb{C}^n,dV)$ for some $r>0$ while $T_\\mu^{\\mathrm{ph}}\\notin S_p$, or if any choice of $\\mu$ and $p$ produced a norm ratio outside $[1,2^{\\max\\{1,p\\}}]$. A direct check would compute the singular values of $T_\\mu^{\\mathrm{ph}}$ for a finite weighted sum of point masses and compare the resulting $S_p$ norm with the ball-mass $L^p$ norm.","supporting_citations":[{"cited_title":"Toeplitz operators on pluriharmonic Fock spaces","cited_arxiv_id":"2605.23256","evidence_quote":"Posed the pluriharmonic Schatten conjecture and proved the $p=1$ case, necessity for $p>1$, and sufficiency for radial measures."},{"cited_title":"Isralowitz, J","cited_arxiv_id":null,"evidence_quote":"Supplies the holomorphic Schatten criterion in generalized Fock spaces that Lemma 3.2 imports and normalizes."},{"cited_title":"Isralowitz and K","cited_arxiv_id":null,"evidence_quote":"Gives the one-dimensional holomorphic Fock-space criterion and trace formula that the paper extends to pluriharmonic spaces."},{"cited_title":"Bourin and E.-Y","cited_arxiv_id":null,"evidence_quote":"Provides the finite-dimensional positive block decomposition and symmetric-norm consequences used in Proposition 4.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Rotfel'd's inequality underlies the trace estimates for positive block operators."},{"cited_title":"Fulsche,Toeplitz operators on pluriharmonic function spaces: deformation quan- tization and spectral theory, Integral Equations Operator Theory91(2019), no","cited_arxiv_id":null,"evidence_quote":"Earlier block-matrix decomposition for pluriharmonic function spaces sets the framework used in the splitting."},{"cited_title":"Rozenblum and N","cited_arxiv_id":null,"evidence_quote":"Provides the sesquilinear-form convention for singular Toeplitz symbols that the paper adopts."}],"review_version":1}