{"id":"6aa38a86-7667-4790-81be-12f4eb876ebe","arxiv_id":"1908.07029","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"If a bounded operator A matches a normal operator N in the norm of every polynomial p(A), p(N), then A is normal in the finite-dimensional case and for compact (c,p)-norm case; for the operator norm, this holds exactly when the spectrum of N is a Lavrentieff set.","lead":"An operator that has the same norm for every polynomial in it as a normal operator is usually forced to be normal itself. This paper proves that for finite matrices and for compact operators under (c,p)-norms, and gives a complete necessary and sufficient spectrum condition for the infinite-dimensional operator norm.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.3's reduction to finite-dimensional compressions conflates I_H with R_k, so the claimed polynomial isometry of N_k and A_k is not established.","rationale":"The reader's CONDITIONAL verdict is retained, but for a more serious reason than the rank verification noted in the weakest_assumption. The rank issues around R_k - P_k and R_k - Q_k are real and subordinate. The more load-bearing problem is that the proof of Theorem 4.3 establishes an ambient norm equality involving I_H, then applies Theorem 2.4 on H_k, where the identity must be R_k. The difference is not negligible, and a simple finite-rank example shows ambient equality does not imply the finite-dimensional equality in general. Since this transfer is the only bridge from the Riesz projections to normality of A_k, Theorem 4.3 is not proved by the argument as written. The finite-dimensional Theorem 2.4 and the operator-norm Theorem 3.6 appear independent and sound; the issue is confined to the compact (c,p)-norm theorem. Thus the overall paper remains valuable but requires a corrected Section 4 before its third main claim can be accepted.","tokens_in":12926,"tokens_out":24453,"duration_ms":262613,"concrete_test":"Isolate the transfer lemma the proof needs: for finite-rank S,T supported on H_k, does ||q(0)I_H + S||_{c,p} = ||q(0)I_H + T||_{c,p} for all polynomials q imply ||q(0)R_k + S||_{B(H_k)} = ||q(0)R_k + T||_{B(H_k)}? A direct counterexample to this implication is n=2, c1=c2=1, p=1, H_k=C^2 embedded as the first two coordinates of l2, R_k=I_C2 xor 0, S=diag(-1/2,-1/2), T=0, q(0)=1: the ambient norms are both 2, while on H_k the norms are 1 and 2. This shows the displayed chain cannot justify polynomial isometry of N_k and A_k. The test is to either supply a corrected argument using the Riesz-projection structure to prove the H_k polynomial isometry, or to exhibit a concrete A,N satisfying the hypotheses for which the H_k norms genuinely diverge.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The key step in Theorem 4.3 tries to show that the finite-rank compressions N_k and A_k are polynomially isometric so that Theorem 2.4 applies. The displayed chain proves only an ambient equality: ||q(N_k)||_{c,p} = ||q(0)I + P_k q_1(N)||_{c,p} = ... = ||q(0)I + Q_k q_1(A)||_{c,p}, where q(N_k) is read with the ambient identity I_H. But Theorem 2.4 is then applied after viewing N_k and A_k as operators on H_k; in that space the relevant embedded operator is q(0)R_k + q_1(N_k), not q(0)I_H + q_1(N_k). These differ by q(0)(I_H - R_k), a nonzero operator whose (c,p)-norm is |q(0)|(sum_j c_j)^{1/p}. The following 'Notice also' line merely restates the H_k-version; it does not prove ||q(0)R_k + q_1(N_k)||_{B(H_k)} = ||q(0)R_k + q_1(A_k)||_{B(H_k)}. This is not a harmless technicality: ambient equality of operators of the form aI + S does not force equality after replacing I_H by R_k. For example, with n=2, c1=c2=1, p=1, H_k=C^2, S=diag(-1/2,-1/2) on H_k and T=0, both I_H+S and I_H have first two singular values 1, while on H_k the norms of I_Hk+S and I_Hk are 1 and 2 respectively. Thus the inference used to reach Theorem 2.4 is invalid in general, and the proof provides no additional structure to bridge the gap. The finite-spectrum case inherits the same problem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the question of whether an operator A that is polynomially isometric to a normal operator N with respect to a unitarily-invariant norm must itself be normal. In the finite-dimensional case the authors prove that this is true for every unitarily-invariant norm on M_n(C) (Theorem 2.4). For the operator norm on B(H) they prove a complete dichotomy: if σ(N) is Lavrentieff, every polynomially isometric A is normal, while if σ(N) is not Lavrentieff, a non-normal polynomially isometric operator exists (Theorem 3.6). For compact operators with respect to the (c,p)-norms of Chan, Li and Tu they claim the same automatic normality (Theorem 4.3). A final section shows that if only polynomials without constant term are used, normality can fail, and gives a positive finite-dimensional result for invertible operators.","tokens_in":13293,"tokens_out":33701,"duration_ms":324136,"significance":"If the results hold, the finite-dimensional theorem for all unitarily-invariant norms is a clean extension of earlier Frobenius- and operator-norm results, and the operator-norm dichotomy in terms of Lavrentieff spectra correctly identifies the role of polynomial approximation. The zero-constant-term counterexample in Section 5 is a useful negative result that delineates why the constant term is essential. The C*-algebra argument in Section 3 is short and conceptually appealing. However, the compact (c,p)-norm theorem has a serious proof gap in the reduction to finite-rank compressions, and until this is repaired the paper's main advertised extension to compact operators is not established.","major_comments":[{"comment":"The displayed chain showing that A_k and N_k are polynomially isometric proves only the ambient equality ||q(0)I_H + P_k q_1(N)||_{c,p} = ||q(0)I_H + Q_k q_1(A)||_{c,p} in B(H). The following 'Notice also' replaces I_H by R_k and asserts that the norms of the compressions on H_k agree with these ambient norms, but this replacement is not valid. For an operator S supported on H_k, the ambient operator q(0)I_H + S has infinitely many singular values equal to |q(0)| coming from the infinite-dimensional complement H_k^⊥, whereas the norm on B(H_k) is computed from q(0)I_{H_k} + S alone. When |q(0)| is large enough, the ambient norm is identically |q(0)|(Σ c_j)^{1/p} and carries no information about the finite-rank part. Thus equality of ambient norms does not imply that N_k and A_k are polynomially isometric as operators on H_k, and Theorem 2.4 cannot be applied. The finite-spectrum case, described as 'similar', inherits the same problem. A concrete illustration is H_k = C^2 with c_1 = c_2 = 1 and p = 1: for S = diag(-1/2,-1/2) and T = 0, the ambient operators I_H+S and I_H have the same first two singular values, while the norms restricted to H_k are 1 and 2 respectively.","section":"Section 4.3, proof of Theorem 4.3"},{"comment":"The assertion that ||R_k - P_k||_{c,p} = ||I - P_k||_{c,p} and ||R_k - Q_k||_{c,p} = ||I - Q_k||_{c,p} is 'readily verified' but actually requires that the complements H_k ⊖ Ran P_k and H_k ⊖ Ran Q_k have dimension at least n, so that the first n singular values are all equal to 1. The definition of H_k via α_k = rank P_k + dim(ker Q_k)^⊥ + rank Q_k + 3k guarantees this only for k with 3k ≥ n, and the proof applies the equalities for every k without this rank verification. This is a repairable technical point, but it is load-bearing for the application of Lemma 2.3 and should be stated and proved explicitly.","section":"Section 4.3, definition of H_k"}],"minor_comments":[{"comment":"In the paragraph defining the limit of the P_k, the text writes 'P = sup{Q_k : k ∈ N} = SOT − lim P_k'; this should be 'sup{P_k : k ∈ N}'.","section":"Section 4.3, Case One"},{"comment":"In the chain of inequalities near the end of Case One, the expressions 't_j^{(k)}(A)AA' and 't_j^{(k)}(N)NN' appear to contain redundant letters and should read 't_j^{(k)}(A)A' and 't_j^{(k)}(N)N'.","section":"Section 4.3, final norm estimate"},{"comment":"The definition of the (c,p)-norm does not explicitly state how the singular values s_j(T) are interpreted when dim H < n; the authors should state that the missing singular values are taken to be 0, as is implicitly done in the later use on finite-dimensional spaces.","section":"Section 4.2, Definition of (c,p)-norm"},{"comment":"The step 'Let L = hat{σ(N)}. Then L^o ≠ ∅' is stated without explanation; a sentence justifying why a non-Lavrentieff spectrum forces the polynomially convex hull to have nonempty interior would improve readability.","section":"Section 3.6, proof of (b) implies (a)"}],"recommendation":"major_revision","confidential_remarks":"The finite-dimensional theorem and the operator-norm dichotomy appear sound and publishable, but Theorem 4.3 is advertised as a main result and its proof has a substantial gap. If the authors cannot supply a valid argument showing that the finite-rank compressions are polynomially isometric on H_k, or otherwise prove the compact (c,p) claim, then that theorem should be removed or explicitly reformulated as open. I see no evidence of bad faith; the error is a subtle but genuine confusion between ambient and compressed identities."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The finite-dimensional theorem (2.4) is a clean, self-contained generalization of Gerasimova and Brooks–Condori to any unitarily-invariant norm, and the rank-separating variant is a nice touch. The infinite-dimensional operator-norm result (3.6) is the real prize: the Lavrentieff dichotomy is sharp, the C*-algebra proof is elegant, and the converse construction with the unilateral shift is a neat use of subnormality. Section 5's example about polynomials vanishing at zero is a useful counterpoint.\n\nThe problem is Section 4. The proof of Theorem 4.3 tries to show that the finite-rank compressions N_k and A_k are polynomially isometric so that Theorem 2.4 applies. The displayed chain establishes an equality of (c,p)-norms for the ambient operators q(0)I_H + P_k q_1(N) and q(0)I_H + Q_k q_1(A). But when you view N_k and A_k as operators on H_k, the relevant embedded operator is q(0)R_k + q_1(N_k), not q(0)I_H + q_1(N_k). These differ by q(0)(I_H - R_k), which is nonzero on the complement of H_k. The 'Notice also' line merely restates the H_k-version; it does not connect it to the ambient equality. So the polynomial isometry of N_k and A_k in B(H_k) is not established. This is not a technical quibble: ambient equality of operators of the form aI + S does not survive compression to a subspace, as the concrete 2x2 example shows. The finite-spectrum case inherits the same gap.\n\nElsewhere the paper is in good shape. No circular reasoning, no data issues, and the citation pattern is standard. The self-citation to [8] for the Lavrentieff criterion is appropriate.\n\nVerdict: I would send this to a serious referee. Sections 2 and 3 deserve publication, and Section 4 may be fixable—either by a repaired compression argument or by narrowing the claim. A referee should insist on that repair before acceptance.","headline":"Sections 2 and 3 are strong and publishable, but the compact (c,p)-norm theorem in Section 4 has a genuine proof gap that needs repair.","tokens_in":13822,"tokens_out":7135,"would_cite":true,"duration_ms":66462,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47B15","15A60","15A21"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that if two operators have matching norms for every polynomial image, and one is normal, then the other is forced to be normal in the matrix setting and in the compact-operator (c,p)-norm setting; for the…","keywords":["polynomially isometric","normal operators","unitarily-invariant norm","(c,p)-norm","singular values","Lavrentieff spectrum","operator norm","Riesz projections"],"falsifier":"For a concrete test, take N=diag(λ_1,λ_2,0,...) and A=N+E_12 in a (c,p)-norm; if some polynomial p satisfies ‖p(A)‖_{c,p}=‖p(N)‖_{c,p} while A is nonnormal, Theorem 4.3 is false. Short of that, computing the singular values of R_k−P_k and R_k−Q_k for such a pair at one spectral cutoff settles whether the proof's operative rank assumption is valid.","tokens_in":12708,"feed_emoji":"🎯","tokens_out":8398,"duration_ms":79972,"temperature":0.7,"pith_summary":"The paper asks a sharp question: if A and N have the same norm for every polynomial p(A) and p(N) with respect to a unitarily-invariant norm, and N is known to be normal, must A be normal? The authors prove that in finite-dimensional space the answer is always yes for every unitarily-invariant norm, and when the norm distinguishes projections by rank, A and N are unitarily similar. In infinite-dimensional operator norm the answer is yes exactly when the spectrum of N is a Lavrentieff set; if σ(N) has interior or disconnects the plane, a non-normal polynomially isometric partner is constructed. For compact operators with the (c,p)-norms, normality again transfers without spectral restrictions. The paper also shows that omitting the constant term destroys the conclusion, with a two-dimensional idempotent example.","feed_headline":"Matching all polynomial norms of a normal operator forces normality","feed_subtitle":"A normal operator's polynomial norms rigidly determine normality, with one sharp exception in infinite dimensions.","key_machinery":"The key object is the comparison of Riesz spectral projections P_i=E_N(Δ_i) and Q_i=E_A(Δ_i) at the same spectral sets. Polynomials that isolate individual spectral subsets give ‖P_i‖=‖Q_i‖ and ‖I−P_i‖=‖I−Q_i‖. Lemma 2.3 — if a projection and an idempotent have the same norm, and their complements have the same norm, then the idempotent is a projection — converts this data into selfadjointness of Q_i. In the operator-norm case, the Lavrentieff hypothesis makes the unital algebra generated by N equal to its C*-algebra, so the polynomial isometry extends to a *-isomorphism; when the hypothesis fails, a unilateral shift summand supplies the non-normal partner. In the compact (c,p)-norm case, finite-dimensional cutoffs H_k constructed from these Riesz projections reduce the problem to the matrix theorem.","core_discovery":"The central discovery is that polynomial isometry to a normal operator is a rigid constraint. In M_n(C), for any unitarily-invariant norm, if ‖p(A)‖_u=‖p(N)‖_u for all polynomials and N is normal, then A is normal; if the norm separates projections by rank, A and N are unitarily similar (Theorem 2.4). In B(H) with the operator norm, normality transfers exactly when σ(N) is Lavrentieff; otherwise a non-normal operator B polynomially isometric to N always exists (Theorem 3.6). For compact A and N with N normal, the same conclusion holds under any (c,p)-norm (Theorem 4.3). The proofs identify the polynomial class with the full C*-algebra generated by N in the Lavrentieff case, and use Riesz projections plus a finite-dimensional lemma to force the partner's spectral idempotents to be selfadjoint in the compact case.","pith_inferences":["The finite-dimensional result suggests that in any unitarily invariant norm that separates projections by rank, polynomial isometry to a normal element should be a complete unitary-invariance invariant; one could test this in von Neumann algebra settings where a version of Lemma 2.3 may hold.","The Lavrentieff dichotomy for the operator norm invites the question of whether other unitarily invariant ideals, such as Schatten p-classes with p<∞, have their own spectral-geometry threshold; the Section 4 counterexample with the constant term omitted shows the answer will depend delicately on the role of the identity.","A concrete computational test of Theorem 4.3's 'readily verified' rank assumptions on small compact examples would either close the proof gap or reveal a need for additional hypotheses; that is the natural next step for someone wanting to rely on the compact-operator result."],"forward_implications":["In any finite-dimensional setting, a normal matrix is completely pinned down, up to unitary similarity, by the polynomial norm behaviour of its spectral projections whenever the norm used can tell projections of different ranks apart.","For the operator norm, the line between normality forcing and non-normality allowing is drawn precisely by Lavrentieff spectrum: spectra with interior or with more than one complementary component admit non-normal partners.","The compact (c,p)-norm theorem means that for Schatten-type and Ky Fan type norms, polynomial isometry to a normal compact operator is a genuine normality certificate, even though the analogous statement fails for general bounded operators under the operator norm.","When only polynomials vanishing at zero are compared, the conclusion fails in general; however, for invertible matrices on finite-dimensional spaces, vanishing-at-zero polynomial data still force normality and unitary similarity."],"supporting_citations":[{"why":"Provided the earlier finite-dimensional operator-norm normality criterion that Theorem 2.4 generalises.","marker":"[2]"},{"why":"Introduced the (c,p)-norms whose compact-operator analogue is proved in Theorem 4.3 and supplies the norm-estimate machinery.","marker":"[3]"},{"why":"Supplies the symmetric-gauge inequality used in Lemma 2.2 to compare singular-value sequences.","marker":"[7]"},{"why":"Established the Frobenius-norm version of the matrix result that is extended here to all unitarily invariant norms.","marker":"[11]"},{"why":"The polynomial-approximation result underlying the equivalence between Lavrentieff spectrum and the generation of C*(N) by N.","marker":"[16]"},{"why":"Links unitarily invariant norms to symmetric gauge functions, the basis for Lemma 2.3's norm comparisons.","marker":"[17]"},{"why":"Provides the polynomial approximation theorem used to transfer norm equality from A and N to their Riesz projections.","marker":"[21]"},{"why":"Supplied the projection/idempotent comparison lemma (Lemma 2.3) that forces Riesz idempotents of A to be selfadjoint.","marker":"[22]"}],"fun_headline_variants":["Polynomial isometry to a normal operator forces normality","Matching every polynomial norm of a normal operator implies normality","Polynomial norms of a normal operator: a rigidity test with one exception","If your operator matches a normal one on all polynomials, you're normal"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The compact-operator argument hinges on an unshown rank-counting step: the auxiliary finite-rank projection R_k must differ from the Riesz projections P_k and Q_k in such a way that the first n singular values of each difference are all 1; if that count is wrong, Lemma 2.3 cannot be invoked.","fun_headline_variants_meta":{"raw":{"variants":["Polynomial isometry to a normal operator forces normality","Matching every polynomial norm of a normal operator implies normality","Polynomial norms of a normal operator: a rigidity test with one exception","If your operator matches a normal one on all polynomials, you're normal"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000291,"raw_usage":{"total_tokens":1791,"prompt_tokens":1127,"completion_tokens":664,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":743,"completion_tokens_details":{"reasoning_tokens":592}},"tokens_in":743,"tokens_out":664,"duration_ms":6579,"temperature":1.0,"reasoning_tokens":592,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:32:04.113585+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a concrete test, take N=diag(λ_1,λ_2,0,...) and A=N+E_12 in a (c,p)-norm; if some polynomial p satisfies ‖p(A)‖_{c,p}=‖p(N)‖_{c,p} while A is nonnormal, Theorem 4.3 is false. Short of that, computing the singular values of R_k−P_k and R_k−Q_k for such a pair at one spectral cutoff settles whether the proof's operative rank assumption is valid.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provided the earlier finite-dimensional operator-norm normality criterion that Theorem 2.4 generalises."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced the (c,p)-norms whose compact-operator analogue is proved in Theorem 4.3 and supplies the norm-estimate machinery."},{"cited_title":"Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators , Proc","cited_arxiv_id":null,"evidence_quote":"Supplies the symmetric-gauge inequality used in Lemma 2.2 to compare singular-value sequences."},{"cited_title":"Gerasimova, Unitary similarity to a normal matrix , Linear Algebra Appl","cited_arxiv_id":null,"evidence_quote":"Established the Frobenius-norm version of the matrix result that is extended here to all unitarily invariant norms."},{"cited_title":"Lavrentieff, Sur les fonctions d’une variable complexe repr´ esentables par les series de polynˆ omes, Acta","cited_arxiv_id":null,"evidence_quote":"The polynomial-approximation result underlying the equivalence between Lavrentieff spectrum and the generation of C*(N) by N."},{"cited_title":"Li and N.K Tsing, On the unitarily-invariant norms and some related results , Linear and Multi- linear Algebra 20 (1987), no","cited_arxiv_id":null,"evidence_quote":"Links unitarily invariant norms to symmetric gauge functions, the basis for Lemma 2.3's norm comparisons."},{"cited_title":"Rudin, Real and complex analysis , third ed., McGraw-Hill Book Co., New York, 1987","cited_arxiv_id":null,"evidence_quote":"Provides the polynomial approximation theorem used to transfer norm equality from A and N to their Riesz projections."},{"cited_title":"Viswanath and L.N","cited_arxiv_id":null,"evidence_quote":"Supplied the projection/idempotent comparison lemma (Lemma 2.3) that forces Riesz idempotents of A to be selfadjoint."}],"review_version":1}