{"id":"79dfe0e9-41a7-4984-a1a3-bda6d590ec16","arxiv_id":"2509.11931","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Spectral mapping theorems for point, residual, and bounded approximate point spectra are extended from Banach spaces to strongly continuous locally equicontinuous semigroups on sequentially complete locally convex spaces.","lead":"This paper proves spectral inclusion and mapping theorems for strongly continuous semigroups on Hausdorff locally convex spaces, extending classical Banach space results. It covers point, residual, approximate point, and bounded approximate point spectra, with special attention to periodic semigroups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's verdict was CONDITIONAL based mainly on a minor citation issue and found no load-bearing mathematical error. My stress-test of the central claim found a concrete false statement at t=0 in Theorem 5.3: the proof incorrectly asserts A=0 when t=0. The translation semigroup on L^p(R) satisfies all hypotheses and gives σ_p(A)=∅ but σ_p(T(0))={1}, so the equality as stated fails. This is not a stylistic or citation issue; it changes the theorem statement. However, it is also clearly repairable: all substantive content of the proof addresses t>0, and the classical theorem and Corollary 5.4(b) already use t>0 for the reverse inclusion. Therefore the correct verdict is CONDITIONAL on adding t>0 to the spectral mapping equalities, rather than REJECT. The Carleson-theorem point raised by the reader is not load-bearing: the standard uniqueness argument for Fourier coefficients is sufficient, and Carleson's theorem, while unnecessary, gives a valid route. The weakest assumption identified by the reader (sequential completeness and local equicontinuity) is not where the central claim actually breaks; the break is the t=0 boundary case.","tokens_in":38541,"tokens_out":15111,"duration_ms":160039,"concrete_test":"Test the t=0 case against the translation semigroup on L^p(R): verify that σ_p(A)=∅ for A=d/dx and σ_p(T(0))={1}; then the claimed equality at t=0 gives ∅={1}. This single check settles that Theorem 5.3 cannot be stated for all t≥0. Equivalently, compare with Engel–Nagel IV.3.7, where the reverse point-spectrum inclusion is explicitly stated only for t>0.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim Theorem 5.3 is stated for all t≥0, but the reverse inclusion is false at t=0. In the proof, the case t=0 asserts: 'If t=0, then T(0)=id, A=0 and σ_p(T(0))={1} and σ_p(A)={0}.' This is not true for an arbitrary generator A. A concrete counterexample is the translation semigroup T(t)f(x)=f(x+t) on L^p(R), p<∞. It is strongly continuous and locally equicontinuous (indeed quasi-equicontinuous on a Banach space), with generator A=d/dx. The point spectrum σ_p(A) is empty because an eigenfunction for f'=λf is a multiple of e^{λx}, which is never in L^p(R). Thus e^{0σ_p(A)}=∅, while σ_p(T(0))=σ_p(I)={1} on the nonzero space X. Hence the asserted equality σ_p(T(0))∖{0}=e^{0σ_p(A)} fails. The classical Engel–Nagel theorem states the reverse inclusion only for t>0, and the paper's own Corollary 5.4(b) also restricts to t>0. This is a real, if easily repaired, flaw in the main theorem: the statement should be for t>0 (or the t=0 case handled separately under additional assumptions). The same issue may propagate to Theorem 5.6 and Corollary 5.10/5.15 insofar as they claim t≥0.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops spectral inclusion and spectral mapping theorems for strongly continuous locally equicontinuous semigroups on sequentially complete Hausdorff locally convex spaces. After setting up several notions of spectrum (point, residual, approximate, bounded approximate, topological), it proves inclusion theorems (Theorem 5.1), a point-spectrum spectral mapping theorem (Theorem 5.3), a residual-spectrum mapping theorem (Theorem 5.6), and mapping theorems for bounded (sequential) approximate point spectra under Schwartz-type or eventual-uniform-continuity assumptions (Corollaries 5.10 and 5.15). A substantial part of the paper (Section 4) is devoted to periodic semigroups, giving spectral characterizations and Fourier-type decompositions. The proofs are detailed and closely follow the classical Banach-space arguments of Engel--Nagel and van Neerven, adapted to the locally convex setting.","tokens_in":38777,"tokens_out":11968,"duration_ms":132490,"significance":"If the stated results are corrected to the appropriate time range, this is a substantial and useful extension of classical semigroup spectral theory. The paper carefully treats pathologies of locally convex spaces, provides explicit examples (e.g., Hardy-space composition semigroups), and gives complete proofs of the inclusion and mapping theorems. The central arguments are derived from established external results (Komura, Albanese--Bonet--Ricker, Engel--Nagel, van Neerven), and the author's own prior work is used only for supporting examples and technical facts; I see no circularity. The most valuable elements are the point-spectrum mapping theorem, the periodic-semigroup analysis, and the residual-spectrum theorem, all of which genuinely go beyond the Banach-space setting.","major_comments":[{"comment":"The statement claims equality for all t≥0, but it is false at t=0. The proof says: 'If t=0, then T(0)=id, A=0 and σ_p(T(0))={1} and σ_p(A)={0}.' However, A is not 0 for a general generator. For example, the translation semigroup T(t)f(x)=f(x+t) on L^p(R), 1≤p<∞, is strongly continuous and locally equicontinuous, and its generator A=d/dx has empty point spectrum because no nonzero L^p function satisfies f'=λf. Thus σ_p(T(0))\\{0}={1} while e^{0σ_p(A)}=e^∅=∅. The theorem should be stated for t>0, as in the classical Engel--Nagel result and as the paper itself does in Corollary 5.4(b). This is a load-bearing error in the main theorem's statement, not merely a typo.","section":"Theorem 5.3"},{"comment":"The asserted identity ker(λ-A)=∩_{t≥0} ker(e^{λt}-T(t)) is false as written. Since T(0)=id and e^{λ·0}=1, the t=0 term in the intersection is ker(1-id)=X whenever X≠{0}. Hence for any λ∉σ_p(A), the right-hand side equals X, while the left-hand side is {0}. The translation semigroup on L^p(R) with λ=0 gives a concrete counterexample: A=d/dx has trivial kernel, but ∩_{t≥0} ker(e^{0·t}-T(t)) = X. The correct identity is with t>0. This is a direct consequence of the same t=0 oversight as in Theorem 5.3.","section":"Corollary 5.4(a)"},{"comment":"The t≥0 claims in these results are either inherited from the flawed Theorem 5.3 or are separately unjustified. Theorem 5.6 applies Theorem 5.3 to the dual semigroup on X^⊙, so the t=0 failure carries over. For the translation semigroup on L^p(R), σ_r(A)=∅ by Proposition 3.9(c) because the adjoint generator has no eigenvalues, while σ_r(T(0))=σ_r(I)∋1; hence σ_r(T(0))\\{0}≠e^{0σ_r(A)}. For Corollary 5.10, Proposition 5.8 is invoked, but its proof reduces to t=1 by rescaling with c=t, so t=0 is not actually covered. Corollary 5.15 treats t=0 by asserting that 'both sides are equal to {1}', which presupposes σ_seq_bap(A) is nonempty; this requires proof. The safe and consistent fix is to state all these spectral mapping theorems for t>0.","section":"Theorem 5.6 and Corollaries 5.10, 5.15"}],"minor_comments":[{"comment":"The invocation of Carleson's theorem is unnecessary and misleading. In Theorem 4.9, the functions involved are continuous and 2π-periodic; equality of their Fourier coefficients implies equality by Fejér's theorem (uniform convergence of Cesàro means), not by Carleson's theorem. Similarly, in Proposition 5.8 the statement that a nonzero continuous periodic function has a nonzero Fourier coefficient is elementary and follows from Fejér's theorem. Using Carleson here attributes a deep result where a standard one suffices.","section":"Theorem 4.9 and Proposition 5.8"},{"comment":"The proof begins 'Due to Theorem 5.1 (c) and (h)...'; Theorem 5.1(h) concerns the residual spectrum and is irrelevant to the point-spectrum equality. Only Theorem 5.1(c) is needed for the inclusion direction.","section":"Theorem 5.3 proof"},{"comment":"There is a stray textual artifact 'it:quasi-equi' in the line following the citation for Definition 2.2. Also, in Example 4.7, 'holmorphic' should be 'holomorphic'.","section":"Definition 2.2"},{"comment":"Even if the t>0 restriction is adopted, the t=0 sentence 'both sides ... are equal to {1}' should either be removed or justified: it requires knowing that 0 lies in σ_seq_bap(A), which is not automatic for arbitrary generators.","section":"Corollary 5.15, t=0 case"}],"recommendation":"major_revision","confidential_remarks":"The t=0 problem is confined to statements and not to the main t>0 machinery; the proofs for t>0 appear sound, so this is fixable within the manuscript's scope. I recommend asking for a revision that restricts the mapping theorems to t>0 (or proves the t=0 cases under explicit additional assumptions) and corrects Corollary 5.4(a). The paper's central contribution is valuable and should be published after these fixes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on Kruse's paper. The core content is a genuine and useful extension of the classical spectral inclusion and mapping theorems of Engel–Nagel and van Neerven to strongly continuous locally equicontinuous semigroups on sequentially complete Hausdorff locally convex spaces. The proofs are detailed and follow the Banach-space arguments carefully, adapting them with the right locally convex tools (Riemann integrals, equicontinuity, dual semigroup results from Komura and Albanese–Bonet–Ricker). The spectral inclusion theorem (Thm 5.1) and the point-spectrum mapping theorem (Thm 5.3) for t>0 look correct, and the residual-spectrum theorem (Thm 5.6) and the bounded approximate point spectrum results under extra assumptions are valuable additions. I found no load-bearing gap in the central arguments.\n\nThat said, there is a real bug at t=0. Theorem 5.3 states the equality σ_p(T(t))\\{0}=e^{tσ_p(A)} for all t≥0. The proof for t=0 asserts A=0, which is false for a general generator. If σ_p(A)=∅ (e.g., the translation semigroup on L^p(R)), then e^{0σ_p(A)}=∅, but σ_p(T(0))\\{0}={1} for nonzero X. So the statement is false at t=0. The fix is easy: state the theorem for t>0, or treat t=0 as a separate trivial case with the obvious caveat. The same issue propagates to Theorem 5.6 and Corollaries 5.10/5.15. The paper's own Corollary 5.4(b) already restricts t>0, so the authors nearly had it right.\n\nA second soft spot: Carleson's theorem is invoked in Theorem 4.9 and Proposition 5.8 to conclude that a continuous function with zero Fourier coefficients is zero. That's unnecessary and misattributes a deep theorem; Fejér's theorem (or even elementary uniqueness) is the right tool. This is minor but should be corrected.\n\nThe self-citations are not a problem. They are for supporting examples and technical facts, not for the main results.\n\nOverall: the paper is a solid piece of work, worth publishing after a revision. The t=0 statement must be corrected and the Carleson references replaced. I would send it to peer review.","headline":"Solid extension of classical spectral mapping theorems to locally convex spaces, but the main theorem has a small t=0 bug that needs fixing; worth a revision.","tokens_in":39315,"tokens_out":3092,"would_cite":true,"duration_ms":32791,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47A10","47A25","47D06","46A70"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for strongly continuous locally equicontinuous semigroups on sequentially complete Hausdorff locally convex spaces, the nonzero point spectrum of each semigroup operator T(t) is exactly the exponential of the generator","keywords":["spectral mapping theorem","spectral inclusion theorem","strongly continuous semigroup","locally convex space","point spectrum","residual spectrum","bounded approximate point spectrum","periodic semigroup"],"falsifier":"Find a strongly continuous locally equicontinuous semigroup on a sequentially complete Hausdorff locally convex space where, for some $t>0$, an eigenvalue $\\lambda\\neq 0$ of $T(t)$ is not equal to $e^{\\mu t}$ for any $\\mu$ in the point spectrum of the generator $A$; such a case would disprove Theorem 5.3.","tokens_in":38332,"feed_emoji":"📐","tokens_out":5210,"duration_ms":53832,"temperature":0.7,"texified_at":"2026-08-05T20:30:29.563324+00:00","pith_summary":"The paper extends the classical spectral mapping theorem for strongly continuous semigroups from Banach spaces to Hausdorff locally convex spaces. Its central result, Theorem 5.3, shows that for a strongly continuous locally equicontinuous semigroup on a sequentially complete Hausdorff locally convex space, the nonzero point spectrum of $T(t)$ equals $e^{t\\sigma_p(A)}$ for every $t\\ge 0$. The same framework yields spectral inclusion theorems for several spectra, a spectral mapping theorem for the residual spectrum under additional completeness assumptions, and spectral mapping theorems for bounded approximate point spectra on generalised Schwartz spaces or for eventually uniformly continuous semigroups. This matters because many natural function spaces are locally convex but not Banach, and the result lets one read spectral properties of a semigroup from those of its generator.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":6331,"prompt_tokens":751,"completion_tokens":5580,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":751,"completion_tokens_details":{"reasoning_tokens":4867}},"feed_headline":"Point spectrum formula proven for locally convex semigroups","feed_subtitle":"Banach-space spectral mapping identity now proven for locally convex semigroups, with residual and approximate spectra included.","key_machinery":"The load-bearing identity is the semigroup integral identity of Proposition 2.4: for every $\\lambda$ and $t$, $e^{-\\lambda t}T(t)x - x = (A-\\lambda)\\int_0^t e^{-\\lambda s}T(s)x\\,ds$ for $x\\in X$, with a companion identity for $x\\in D(A)$. These identities, together with the uniform bound (18) supplied by local equicontinuity, convert spectral properties of $A$ into spectral properties of $T(t)$. The reverse direction uses the periodic semigroup analysis of Section 4, which gives a Laurent-series description of the resolvent and a characterisation of periodic semigroups whose generator has point spectrum in $2\\pi i/\\rho \\mathbb{Z}$ and eigenvectors spanning a dense subspace.","core_discovery":"Theorem 5.3 states: if $(T(t))_{t\\ge 0}$ is a strongly continuous locally equicontinuous semigroup on a sequentially complete Hausdorff locally convex space $X$ with generator $(A,D(A))$, then $\\sigma_p(T(t))\\setminus\\{0\\} = e^{t\\sigma_p(A)}$ for all $t\\ge 0$. The proof rescales the semigroup to reduce to the eigenvalue 1, restricts to the eigenspace $\\ker(1-T(1))$, and uses a characterisation of periodic semigroups (Theorem 4.8) that forces the generator's point spectrum on this subspace to lie in a discrete imaginary lattice, so the eigenvalue 1 must be the exponential of a point-spectral value of $A$. The paper also proves inclusion theorems for point, residual, approximate, bounded approximate, and $\\sigma_*$ spectra (Theorem 5.1), a","pith_inferences":["The eigenspace decomposition in Corollary 5.4 suggests that, in the locally convex setting, eigenvalues of T(t) can be studied through the countable family of generator eigenvalues differing by multiples of 2πi/t, potentially leading to multiplicity formulas analogous to the Banach-space theory.","Because many Fréchet spaces automatically make strongly continuous semigroups locally equicontinuous, Theorem 5.3 should apply directly to a broad class of evolution equations on spaces of smooth functions, even when quasi-equicontinuity fails.","One testable extension is whether the residual-spectrum theorem's extra assumptions (sequential completeness of the strong dual and nonemptiness of the algebraic resolvent set) can be dropped or relaxed, since the point-spectrum theorem does not need them.","The periodic-semigroup characterisation gives a constructive route to examples: any locally convex space supporting a semigroup whose generator has point spectrum in a discrete imaginary lattice and dense eigenvectors must produce a periodic semigroup, which may help build explicit counterexamples for other spectral questions."],"forward_implications":["For every t≥0, the eigenspaces are related by ker(λ−A)=⋂_{t≥0} ker(e^{λt}−T(t)) and ker(e^{λt}−T(t))=span(⋃_{n∈ℤ} ker(λ+2πin/t−A)) for t>0 (Corollary 5.4).","Under the extra assumptions that the strong dual is sequentially complete and the algebraic resolvent set of A is nonempty, the residual spectrum satisfies σ_r(T(t))\\{0}=e^{tσ_r(A)} for all t≥0 (Theorem 5.6).","On sequentially complete generalised Schwartz spaces, the bounded and sequential bounded approximate point spectra satisfy the spectral mapping theorem, both in net and sequence form (Corollary 5.10).","For eventually uniformly continuous locally equicontinuous semigroups on sequentially complete spaces, the sequential bounded approximate point spectrum maps exponentially (Corollary 5.15).","Periodic semigroups are characterised spectrally: a strongly continuous locally equicontinuous semigroup is periodic if and only if σ_*(A)=σ_p(A)⊆2πiαℤ for some α>0 and the eigenvectors span a dense subspace (Theorem 4.8)."],"fun_headline_variants":["Point spectrum formula proven for locally convex semigroups","Semigroup spectral mapping extends to locally convex spaces","Point spectrum identity holds for locally convex semigroups","Spectral mapping theorem for locally convex semigroups","Locally convex semigroups: point spectrum formula proven"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is joint: the space must be sequentially complete and the semigroup locally equicontinuous, since the integral identities and the uniform bound used at every step fail without both.","fun_headline_variants_meta":{"raw":{"variants":["Point spectrum formula proven for locally convex semigroups","Semigroup spectral mapping extends to locally convex spaces","Point spectrum identity holds for locally convex semigroups","Spectral mapping theorem for locally convex semigroups","Locally convex semigroups: point spectrum formula proven"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000239,"raw_usage":{"total_tokens":1272,"prompt_tokens":584,"completion_tokens":688,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":328,"completion_tokens_details":{"reasoning_tokens":612}},"tokens_in":328,"tokens_out":688,"duration_ms":6690,"temperature":1.0,"reasoning_tokens":612,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T16:40:08.117225+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a strongly continuous locally equicontinuous semigroup on a sequentially complete Hausdorff locally convex space where, for some $t>0$, an eigenvalue $\\lambda\\neq 0$ of $T(t)$ is not equal to $e^{\\mu t}$ for any $\\mu$ in the point spectrum of the generator $A$; such a case would disprove Theorem 5.3.","supporting_citations":[],"review_version":1}