{"id":"f8396361-a771-40a2-835d-e2a4b0802441","arxiv_id":"2412.13165","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For self-adjoint operators with 0 in the essential spectrum, the unitary, isometric, and spectral distances on different Hilbert spaces coincide; the quasi-unitary distance is equivalent to them up to a universal constant.","lead":"This paper defines and compares three distances between operators acting on different Hilbert spaces: unitary, isometric, and quasi-unitary. The main theorems show that, for self-adjoint operators with 0 in the essential spectrum, the first two coincide and the third is equivalent up to a universal constant, giving approximation theory a common metric language.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem C's upper-bound proof uses the wrong off-diagonal norm identity; for non-normal R2 the expression in Eq. (5.7) is not bounded by δJ, leaving the general bounded-operator claim unproved as written.","rationale":"The reader identified the correct locus of concern: the upper-bound proof of Theorem C, specifically Eq. (5.7), conflates two different norm expressions for non-normal R2. This is a genuine gap in the proof as written, and it invalidates the claimed general bounded-operator version of Theorem C until repaired or the statement is restricted. However, the reader's explicit counterexample is numerically incorrect: for R1 = 1, R2 = [[0,1],[0,0]], and J = (1/√2,0), the fourth term ‖JR1*−R2*J‖ equals 1, so δJ = 1 rather than 1/√2, and the constructed embedding has norm ≈ 1.31 ≤ √3. The same distinction between ‖R2*(id−JJ*)R2‖ and ‖R2(id−JJ*)R2*‖ persists with a corrected example, so the concern about Eq. (5.7) stands independently. The self-adjoint case, where the two quantities coincide, is not affected; Theorem B, Corollary D, and the self-adjoint parts of Theorem F rest on different arguments and appear sound. Therefore the appropriate verdict remains CONDITIONAL: the paper's main applications are plausible, but the general non-self-adjoint form of Theorem C lacks a valid proof in the text. My recommendation is UNCHANGED relative to the reader's verdict because my review neither clears the gap nor widens it beyond the conditional status already assigned.","tokens_in":37048,"tokens_out":22098,"duration_ms":188007,"concrete_test":"Recompute Eq. (5.7) using the correct identity (5.3j) for ‖P2DP1⊥‖². For the explicit R1 = 0, R2 = [[0,1],[0,0]], and J = (1/√2)|e1⟩⟨g1|, verify that ‖R2(id−JJ*)R2*‖ = 1 while the δJ-controlled quantity ‖R2*(id−JJ*)R2‖ = 1/2, and then check whether the constructed operator D satisfies ‖D‖ ≤ √3 δJ for this J. If it does, test whether the proof can be repaired by replacing J with a minimizer whose polar part is aligned with the ranges of R1 and R2, or by adding the fourth δJ term; if no repair preserves the constant √3 for all bounded R2, Theorem C should be restricted to self-adjoint (or normal) operators.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the second inequality in Theorem C, proved in Section 5 via Eq. (5.7). To control the block P2DP1⊥, the proof bounds it by ‖R2*(id−JJ*)R2‖. But Proposition 5.4 gives the norm of P2DP1⊥ as ‖P2DP1⊥‖² = ‖P1⊥D*P2‖² = ‖R2(id−JJ*)R2*‖ (Eq. (5.3j)), while ‖R2*(id−JJ*)R2‖ is the square of the norm of the different operator P2D*P1⊥. For self-adjoint R2 the two expressions coincide, but for general bounded R2 they differ. A concrete witness is R1 = 0 on C², R2 = [[0,1],[0,0]] on C², and J = (1/√2)|e1⟩⟨g1| in an appropriate basis. Here ‖R2*(id−JJ*)R2‖ = 1/2 while ‖R2(id−JJ*)R2*‖ = 1, so the proof under-estimates the required block norm by a factor of 2. The reader's published counterexample (R1 = 1, R2 nilpotent, J = (1/√2,0)) does not land as stated because it omits the fourth term in δJ: ‖JR1*−R2*J‖ = 1, so δJ = 1, not 1/√2. The underlying defect is nevertheless real: Eq. (5.7) asserts an estimate that is not a consequence of the identities proved in Proposition 5.4. The theorem may still be true, and the self-adjoint applications in Corollary D and Theorem F are unaffected, but the proof as written does not establish the general non-self-adjoint upper bound.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces and compares three distances between bounded operators acting on different separable Hilbert spaces: the unitary distance duni, the isometric distance diso (after Weidmann), and a quasi-unitary distance dque with sandwiched defect terms. The main theorems assert that for self-adjoint operators with 0 in the essential spectrum, duni equals diso and both equal the spectral distance dspec (Theorems A and B, Corollary D); that for arbitrary bounded operators, dque and diso are equivalent up to the universal constant sqrt(3) (Theorem C); and that the associated convergence notions for resolvents—Weidmann convergence, QUE-convergence, and convergence in these distances—coincide with matching speeds (Theorem F). The proofs rely on crude multiplicity functions and the Azoff–Davis/Davidson equality for unitary orbits, plus explicit embedding constructions for the isometric and quasi-unitary distances.","tokens_in":37186,"tokens_out":21317,"duration_ms":174302,"significance":"If established, the framework gives a coherent quantitative language for spectral approximation of operators on varying Hilbert spaces, unifying earlier convergence notions and showing that spectral, unitary, isometric, and quasi-unitary distances carry the same information (up to a constant). The paper is careful in attributing the unitary-orbit results to Azoff–Davis and Davidson, and the self-adjoint equality Theorem B appears sound and is proven with a clean crude-multiplicity argument. The self-adjoint consequences (Corollary D, Theorem A, and the self-adjoint parts of Theorem F) are valuable. However, the general bounded-operator part of Theorem C is not proven as written, and this gap propagates into the non-self-adjoint statements of Corollary E and Theorem F.","major_comments":[{"comment":"The upper-bound proof of Theorem C for general bounded operators is not supported by Proposition 5.4. The estimate in Eq. (5.7) bounds ‖P2DP⊥1‖² by ‖R2*(id−JJ*)R2‖, but the identity proved in the paper gives ‖P2DP⊥1‖² = ‖R2(id−JJ*)R2*‖ (Eq. (5.3j)); Eq. (5.3h), which concerns the different block P⊥1DP2, is itself misstated for non-self-adjoint R2 (the correct right-hand side is ‖R2*(id−JJ*)R2‖). These two numbers differ in general: for R2 = [[0,1],[0,0]] and J = (1/√2)(1,0) one has ‖R2*(id−JJ*)R2‖ = 1/2 but ‖R2(id−JJ*)R2*‖ = 1. Since δJ controls only the former, the chain (2δ²+δ²) in (5.7) is unjustified for non-self-adjoint operators. The self-adjoint case is unaffected, and the first inequality dque ≤ diso can be repaired using Eq. (5.3e), but as written Theorem C is not established in the generality stated.","section":"Section 5, Theorem C proof, Eq. (5.7)"},{"comment":"There is a mismatch between the QUE-convergence conditions and the quasi-unitary distance used in the equivalence. Definition 7.2 imposes estimates on ‖Rn(id−J*J)Rn‖ and ‖R∞(id−JJ*)R∞‖ (no adjoints on the resolvents), while the distance dque in Eq. (1.10b) uses ‖R1*(id−J*J)R1‖ and ‖R2*(id−JJ*)R2‖. For non-self-adjoint resolvents these are not the same expressions. Consequently the asserted equivalence (c)⇔(d) in Theorem F for general closed operators does not follow from Theorem C without an additional argument or a revision of Definition 7.2. The self-adjoint part of Theorem F, where the resolvents are self-adjoint, is not affected.","section":"Section 7, Definition 7.2 and Theorem F"}],"minor_comments":[{"comment":"The computation of δJ(R,0) drops the fourth term ‖JR*‖ from Eq. (1.10b); the stated value ‖R‖/√2 is still correct (one may take J = (1/√2)U with U unitary and use ‖SR‖²+‖CR‖² = ‖R‖²), but the proof should account for this term explicitly.","section":"Section 6.2, Proposition 6.3"},{"comment":"Eq. (5.3h) should be corrected to ‖P⊥1DP2‖² = ‖R2*(id−JJ*)R2‖; this typo is related to the gap in Theorem C and should be fixed even if the main theorem is ultimately repaired by a different argument.","section":"Section 5, Proposition 5.4"},{"comment":"The notation in the definition of ddisc is hard to read because the spectral projection symbol appears as /BD; the authors should ensure the typeset version uses a clear projection notation.","section":"Section 1.1, Eq. (1.4)"}],"recommendation":"major_revision","confidential_remarks":"The reader's stress-test concern lands: the proof of the second inequality in Theorem C uses the wrong off-diagonal norm identity. I do not regard this as a disproof of Theorem C, but the general bounded-operator claim is currently unsupported. Since the self-adjoint core of the paper is sound and in my view publishable, revision should either repair the general case or restrict the affected statements to self-adjoint operators. The mismatch in Definition 7.2 should also be addressed before the non-self-adjoint parts of Theorem F are claimed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThis paper deserves a serious referee. The main contribution is a genuinely useful unification of distances between operators acting on different Hilbert spaces: unitary orbit distance, isometric distance (Weidmann), and the authors' quasi-unitary distance. Theorem B — equality of unitary and isometric distance for self-adjoint operators when 0 lies in both essential spectra — is new and cleanly proved via crude multiplicity functions, with honest attribution to Azoff-Davis and Davidson for the ingredients. Theorem C's two-sided comparison with explicit constant sqrt(3) is also new, and the applications to resolvent convergence and spectral distance look valuable. The paper is careful about attribution and openly flags open questions (sharp constant, triangle inequality).\n\nThe soft spot is in the proof of the second inequality of Theorem C (diso ≤ sqrt(3) dque) for general bounded operators. The proof uses Proposition 5.4 to bound the off-diagonal block P2DP1⊥, but Eq. (5.3j) gives ||P2DP1⊥||² = ||R2(id - JJ*)R2*||, while the argument needs a bound in terms of ||R2*(id - JJ*)R2||. These coincide for self-adjoint R2 but differ for non-normal operators. The reader's proposed counterexample has a mistake: for R1 = 1, R2 nilpotent, J = (1/sqrt(2), 0), the term ||JR1* - R2*J|| equals 1, so δJ = 1, not 1/sqrt(2). But the underlying concern survives: Eq. (5.7) asserts an estimate that does not follow from the identities proved. The general bounded-operator claim is therefore unproved as written. The self-adjoint special case — the one used in Corollary D and Theorem F — appears sound and unaffected.\n\nOther soft spots are minor by comparison. The example section is informative, though some computations rely on Halmos' two-projections theorem without full derivation. The generalized triangle inequality for dque is proved using Theorem C, which is structurally a bit awkward but acceptable after Theorem C is established.\n\nWho gets value: anyone working on norm-resolvent convergence, graph approximations of manifolds or fractals, or spectral convergence for operators on varying spaces. The self-adjoint part gives a clean metric framework with sharp equivalence of spectral convergence notions. I'd cite the paper for the self-adjoint results even while cautioning about the non-self-adjoint statement.\n\nRecommendation: send to peer review. The referee should require the authors to repair or restrict Theorem C. If the general bounded claim cannot be fixed with a corrected norm estimate, the statement should be narrowed to self-adjoint (or normal) operators, where the proof works. That is a substantial but local revision.\n\nYours,","headline":"Solid self-adjoint results, but the general version of Theorem C rests on a bad norm identity that the authors need to repair or retract.","tokens_in":37981,"tokens_out":2024,"would_cite":true,"duration_ms":19465,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47A10","47A58","47B15","47A55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that three natural distances between operators on different Hilbert spaces are quantitatively equivalent, with unitary and isometric distances exactly equal for self-adjoint operators whose essential spectrum contains 0.","keywords":["unitary distance","isometric distance","quasi-unitary equivalence","crude multiplicity functions","Lévy-Prokhorov distance","spectral distance","operator convergence","varying Hilbert spaces"],"falsifier":"For $R_1=1$ on $\\mathbb{C}$, $R_2=\\left(\\begin{smallmatrix}0&1\\\\0&0\\end{smallmatrix}\\right)$ on $\\mathbb{C}^2$, and $J=\\mathrm{diag}(1/\\sqrt{2},0)$, compute the two numbers $\\|R_2^*(\\mathrm{id}-JJ^*)R_2\\|$ and $\\|R_2(\\mathrm{id}-JJ^*)R_2^*\\|$; they come out as $1/2$ and $1$, respectively, so the norm identity used in the proof of Theorem C fails. Computing $d_{\\mathrm{iso}}$ and $d_{\\mathrm{que}}$ exactly for this pair then settles whether the universal $\\sqrt{3}$ bound itself survives for non-normal operators or needs an adjoint-adjusted distance.","tokens_in":36569,"feed_emoji":"📏","tokens_out":16331,"duration_ms":129010,"temperature":0.7,"pith_summary":"Operators in applications often live on different Hilbert spaces—think of a Laplacian on a fractal approximated by discrete graph Laplacians, or a manifold squeezed to a graph. This paper shows that three natural ways of measuring how close two such operators are carry the same quantitative information. For self-adjoint operators with 0 in the essential spectrum, the distance between unitary orbits and the distance obtained by embedding both spaces isometrically into a common space are exactly equal, and both equal a spectral distance built from Hausdorff distance plus multiplicity-aware counting of discrete eigenvalues. For arbitrary bounded operators, a third distance based on quasi-unitary identification operators is equivalent to the isometric distance up to the universal factor $\\sqrt{3}$. Consequently, convergence of operators on varying spaces can be measured by any of these metrics with the same convergence speed, and spectral convergence becomes a special case of metric convergence.","feed_headline":"Operator distances on changing Hilbert spaces collapse into one","feed_subtitle":"Unitary, isometric, and quasi-unitary metrics agree up to a universal constant, unifying convergence on varying spaces.","key_machinery":"The load-bearing objects are the crude multiplicity function $\\alpha_R(\\lambda)=\\lim_{\\varepsilon\\to 0}\\operatorname{rank} \\mathbf{1}_{(\\lambda-\\varepsilon,\\lambda+\\varepsilon)}(R)$, which records how many spectral directions accumulate at each point, together with the Lévy-Prokhorov distance between such functions; the isometric embedding picture with difference operator $D=\\iota_1R_1\\iota_1^* - \\iota_2R_2\\iota_2^*$; and the quasi-unitary identification operator $J:H_1\\to H_2$ with defect terms $\\|R_1^*(\\mathrm{id}-J^*J)R_1\\|^{1/2}$, $\\|R_2^*(\\mathrm{id}-JJ^*)R_2\\|^{1/2}$, $\\|JR_1-R_2J\\|$, and $\\|JR_1^*-R_2^*J\\|$. The proof of Theorem B uses invariance of $\\alpha_R$ under isometries when $0$ is in the essential spectrum; the proof of Theorem C constructs a parent space $H_1\\oplus H_2$, sets $\\iota_1=((\\mathrm{id}-J^*J)^{1/2},J)$, $\\iota_2=(0,\\mathrm{id})$, and bounds $D$ by decomposing it as $P_2DP_1+P_2DP_1^\\perp+P_2^\\perp DP_1$ with orthogonal projections onto the ranges of the isometries, yielding the constant $\\sqrt{3}$ from Pythagoras.","core_discovery":"The paper's central claim is that the choice of how to compare operators on different Hilbert spaces does not matter quantitatively. Concretely, if $R_1$ and $R_2$ are bounded self-adjoint operators and $0$ lies in the essential spectrum of both, then $d_{\\mathrm{uni}}(R_1,R_2)=d_{\\mathrm{iso}}(R_1,R_2)=d_{\\mathrm{spec}}(R_1,R_2)$; and for any bounded operators on separable Hilbert spaces, $d_{\\mathrm{que}}(R_1,R_2)\\le d_{\\mathrm{iso}}(R_1,R_2)\\le \\sqrt{3}\\,d_{\\mathrm{que}}(R_1,R_2)$. The equality is driven by the fact that $0$ in the essential spectrum makes the crude multiplicity function invariant under isometric embedding, so unitary-orbit data survive the embedding. The equivalence with the quasi-unitary distance is proved by constructing an explicit parent space $H_1\\oplus H_2$ with an embedding built from the defect operator of the identification $J$, and estimating the difference operator by a Pythagorean split of its three off-diagonal pieces.","pith_inferences":["The constant $\\sqrt{3}$ in Theorem C is probably not optimal: the paper itself exhibits a lower bound of $\\sqrt{2}$, and tightening the gap between $\\sqrt{2}$ and $\\sqrt{3}$ would sharpen all convergence-speed statements for the quasi-unitary distance.","Because the proof of the bounded-operator upper bound compares two quadratic defect expressions that need not agree for non-normal operators, a natural repair is to redefine or adjust the quasi-unitary distance by placing adjoints symmetrically; the self-adjoint applications in spectral geometry and fractal approximation would be unaffected.","The metric unification suggests a practical recipe for numerical analysis on varying spaces: pick whichever of the three distances is easiest to compute in a given problem, and the resulting convergence criterion (and even its order) transfers to the others.","The isometric distance is formally a Gromov-Hausdorff-type distance for graphs of operators with isometries constrained to diagonal form; exploring that connection could yield a Banach-space analogue or a link to the classical gap distance between subspaces."],"forward_implications":["For self-adjoint operators with $0$ in the essential spectrum, the three metrics $d_{\\mathrm{uni}}$, $d_{\\mathrm{iso}}$, and $d_{\\mathrm{spec}}$ give identical distances; convergence in any one of them is convergence in all, with the same speed.","For any bounded pair, $d_{\\mathrm{que}}$ and $d_{\\mathrm{iso}}$ are equivalent up to the universal constant $\\sqrt{3}$, so quasi-unitary convergence and generalized norm resolvent convergence are equivalent even across different parent spaces.","The unitary distance $d_{\\mathrm{uni}}(R_1,R_2)$ always bounds the Hausdorff distance of the spectra; when both operators have purely essential spectrum, the two distances are exactly equal.","If $0$ is missing from an essential spectrum, the equality $d_{\\mathrm{uni}}=d_{\\mathrm{iso}}$ can fail, as the counterexamples in Section 6 show, so the spectral condition is genuinely needed.","The quotient of self-adjoint operators with $0$ in the essential spectrum by approximate unitary equivalence is a Hausdorff metric space isometric to the space of crude multiplicity functions with the Lévy-Prokhorov distance."],"supporting_citations":[{"why":"Proves the unitary distance equals the Lévy-Prokhorov distance of crude multiplicity functions, the backbone of Theorem B.","marker":"[AD84]"},{"why":"Introduced crude multiplicity functions and characterised the closure of unitary orbits, supplying the invariant behind Theorem A.","marker":"[GP74]"},{"why":"Extends unitary-orbit distance results to normal operators and gives spectral-distance equalities used in Theorem A.","marker":"[D86]"},{"why":"Previous article by the authors defining quasi-unitary and Weidmann-type convergence, which Theorem F complements and refines.","marker":"[PZ22]"},{"why":"Weidmann's generalized norm resolvent convergence is the source of the isometric distance concept.","marker":"[W00]"},{"why":"Introduced the quasi-unitary distance, whose open question this paper answers.","marker":"[PS20a]"},{"why":"The parent-space construction with defect operators used in the proof of Theorem C is inspired by its techniques.","marker":"[SNFBK10]"},{"why":"Halmos' two-subspaces theorem is used to compute isometric distances in the sharpness examples and counterexamples.","marker":"[Hal69]"}],"fun_headline_variants":["Metrics on distinct Hilbert spaces finally align","One distance to compare operators across spaces","Operator distances unify across Hilbert spaces","When Hilbert spaces differ, distances still agree","A single metric for operators on different spaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The universal factor $\\sqrt{3}$ bound between quasi-unitary and isometric distances for general bounded operators rests on the assumption that the defect of the identification operator has the same size whether the operator sits on the left or on the right of the defect factor; this is automatic for self-adjoint operators but not for arbitrary bounded ones.","fun_headline_variants_meta":{"raw":{"variants":["Metrics on distinct Hilbert spaces finally align","One distance to compare operators across spaces","Operator distances unify across Hilbert spaces","When Hilbert spaces differ, distances still agree","A single metric for operators on different spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000211,"raw_usage":{"total_tokens":1466,"prompt_tokens":1051,"completion_tokens":415,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":667,"completion_tokens_details":{"reasoning_tokens":352}},"tokens_in":667,"tokens_out":415,"duration_ms":4377,"temperature":1.0,"reasoning_tokens":352,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T13:27:53.297663+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $R_1=1$ on $\\mathbb{C}$, $R_2=\\left(\\begin{smallmatrix}0&1\\\\0&0\\end{smallmatrix}\\right)$ on $\\mathbb{C}^2$, and $J=\\mathrm{diag}(1/\\sqrt{2},0)$, compute the two numbers $\\|R_2^*(\\mathrm{id}-JJ^*)R_2\\|$ and $\\|R_2(\\mathrm{id}-JJ^*)R_2^*\\|$; they come out as $1/2$ and $1$, respectively, so the norm identity used in the proof of Theorem C fails. Computing $d_{\\mathrm{iso}}$ and $d_{\\mathrm{que}}$ exactly for this pair then settles whether the universal $\\sqrt{3}$ bound itself survives for non-normal operators or needs an adjoint-adjusted distance.","supporting_citations":[],"review_version":1}