{"id":"dc5cdd64-f9c8-415c-8208-86c997cae508","arxiv_id":"1908.06712","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Operators of Read's type with no non-trivial invariant subset satisfy the Hypercyclicity Criterion, so T⊕T is hypercyclic.","lead":"This note proves that every operator of Read's type with no non-trivial invariant subset has a hypercyclic double T⊕T, hence satisfies the Hypercyclicity Criterion. It confirms that all known counterexamples to the invariant subset problem are compatible with this criterion, leaving one open question narrower.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the proof is a sound consequence of the stated Read-type structural properties.","rationale":"The paper proves a clean conditional result: any operator satisfying the two stated structural properties of Read-type operators and with no non-trivial invariant closed subset satisfies the Hypercyclicity Criterion. The proof is short, and I checked the main steps. The approximation property is extended from |p|≤2 to all polynomials by a doubling argument that is correct up to a harmless off-by-one in the number of iterations. The lay-off interval estimates are correct: for i = floor(c/2) and j = ceil(c/2), the exponents are O(ν/√c), so the norms tend to 1. The vectors w_k and polynomials q_k then satisfy the hypotheses of Proposition 4.2, yielding the criterion. The only subtlety is the claim that a strictly increasing sequence n_k can be chosen with ||T^{c_nk}e0 − 4^k e0||<1. The stated approximation property only gives existence of some n for each k, but because 4^k grows, the corresponding c_n must grow, so one can pass to a subsequence with n_k strictly increasing and relabel the weights 2^{-k}; this is a minor filling-in, not a flaw in the theorem. The reader's weakest-assumption identification of the structural properties from [16] is accurate, but those properties are part of the class definition and are cited, not claimed to be proven here. I therefore find no load-bearing objection and keep the verdict unchanged.","tokens_in":7806,"tokens_out":30072,"duration_ms":291829,"concrete_test":"Verify directly from [16, Sec. 2.2 and 3.1] that for each polynomial p with |p|≤2 and each ε>0, the approximation ||T^{c_n}e0 − p(T)e0||<ε holds for arbitrarily large n, equivalently that the family p_n approximates each p along an unbounded set of indices. This is the one external condition the proof's subsequence argument relies on.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is conditional on two structural properties of Read-type operators taken from [16, Sec. 2.2]: the approximation property of the polynomials p_n, and the extreme largeness of c_n relative to ν_n. These are definitional for the class and supported by the cited constructions, and the proof identifies them explicitly. The doubling argument extending the approximation property from |p|≤2 to all polynomials is arithmetically valid; the lay-off interval estimates give ||T^{i_nk}e0|| and ||T^{j_nk}e0|| tending to 1; and the construction of a strictly increasing sequence n_k follows once one passes to a subsequence, since the target 4^k forces c_{n_k}→∞ and the weights 2^{-k} can be relabelled accordingly. No circularity or internal inconsistency appears. The only residual risk is external: if [16]'s structural properties were not as summarized, the theorem would not apply, but that is outside the paper's argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem 3.2: if T is a bounded operator of 'Read's type' on a separable real or complex Banach space and T has no non-trivial invariant closed subset, then T⊕T is hypercyclic, equivalently T satisfies the Hypercyclicity Criterion. The proof combines two structural properties of Read-type operators taken from [16, Sec. 2.2]—the approximation property of the exponents c_n and the norm behavior on the lay-off intervals—with a sufficient condition for the Hypercyclicity Criterion due to the author [14]. The argument produces sequences w_k and polynomials q_k satisfying the hypotheses of Proposition 4.2, and the paper concludes with remarks on known examples and an open question about hypercyclic vectors of operators that fail the criterion.","tokens_in":7947,"tokens_out":12664,"duration_ms":117993,"significance":"The result is a clean and elegant observation: every known operator without a non-trivial invariant closed subset (all of which are of Read's type) automatically satisfies the Hypercyclicity Criterion, so such operators cannot serve as counterexamples to Question 3.1. The proof is short, explicit, and identifies exactly which structural properties of Read's construction are needed. The paper also draws attention to an interesting question about the size of the set of hypercyclic vectors for operators failing the criterion. The main limitation is the dependence on the cited structural properties from [16] and the sufficient condition from [14], but these are published results and the paper uses them transparently. The stress-test concern about external dependence does not land as a fatal flaw, because the relevant properties are explicitly identified and cited.","major_comments":[],"minor_comments":[{"comment":"The displayed formula for ||e_{i_{n_k}}|| is difficult to parse as typeset; please rewrite it using unambiguous notation and explain how the limit 1 follows from the definition of lay-off intervals in [16, Sec. 2.2].","section":"Section 4, proof of Theorem 3.2"},{"comment":"The existence of a strictly increasing sequence (n_k) with ||T^{c_{n_k}}e0 - 4^k e0|| < 1 is asserted without justification; since for fixed n the vector T^{c_n}e0 cannot satisfy the inequality for all k, one can pass to a subsequence, but this should be stated.","section":"Section 4, proof of Theorem 3.2"},{"comment":"The condition that c_n is 'extremely large' relative to ν_n is not quantified; for the argument it suffices that c_n - ν_n → ∞ and c_n/2 ≥ ν_n + 1 eventually, and this should be stated explicitly.","section":"Section 4, proof of Theorem 3.2"},{"comment":"Proposition 4.2 is stated as a 'rewriting' of Theorem 4.1 but no proof or precise pointer to the statement in [14] is given; please add a reference to the exact result or a short derivation.","section":"Section 4, Proposition 4.2"},{"comment":"The sentence 'all the known examples of operators without non-trivial invariant closed subset do satisfy the Hypercyclicity Criterion' is slightly stronger than what is proved; Theorem 3.2 covers only operators of Read's type, so please qualify the statement accordingly.","section":"Introduction"}],"recommendation":"minor_revision","confidential_remarks":"This is a competent short note. The proof is correct conditional on the structural properties of Read-type operators from [16], and the result is a nice observation. The paper is heavily dependent on the author's prior work, but that is transparent and acceptable. The only issues are presentation-related, so I recommend minor revision rather than acceptance in the current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: Grivaux proves that every operator of Read's type with no non-trivial invariant subset satisfies the Hypercyclicity Criterion, i.e. T⊕T is hypercyclic. This is a clean, modest observation, and the proof is correct. The reader's report is sound; I don't have any material disagreement.\n\nWhat is new: the theorem itself. It is not in the literature, and it applies to all the known counterexamples to the invariant-subset problem: Read's operators, Grivaux–Roginskaya's constructions, and the quasinilpotent operators of Gallardo-Gutiérrez–Read. The argument is short: combine a sufficient condition for the Hypercyclicity Criterion (Grivaux's Theorem 4.1, or equivalently the Bès–Peris characterization via the criterion's equivalence with T⊕T hypercyclic) with two structural properties of Read-type operators from [16]—the approximation property of the polynomials p_n in the (c)-working intervals, and the lay-off intervals with c_n extremely large relative to ν_n. The doubling argument that extends the approximation from |p|≤2 to all polynomials is a neat trick and it checks out. The estimates on the lay-off intervals also work: I did the arithmetic and the exponents go to zero exactly as claimed.\n\nSoft spots, in order of importance. First, the class 'of Read's type' is left informal. The paper says it explicitly, and since the proof only uses the two stated properties, this is defensible, but it means the theorem is conditional on properties imported from [16] without being fully restated. A reader who wants to verify the definitions has to go to [16, Sec. 2.2]. Second, the 'extremely large' condition on c_n relative to ν_n is not quantified here; again this is inherited from the construction, and it is a presentation gap rather than a flaw. Third, the significance is modest: Question 3.1 remains open, and no new technique is introduced. That is fine for a note.\n\nThe citation pattern is honest and appropriate. Theorem 4.1 is the author's own earlier result, but it is independent and published; the structural properties are external to this note. No circularity.\n\nThis is a competent, honest note. It does not change the landscape, but it settles the Hypercyclicity Criterion question for every known operator without non-trivial invariant subsets, which was not obvious. It should be peer-reviewed and published. I'd send it to a serious referee, and I'd cite it if I worked in the area.","headline":"Grivaux shows all known Read-type operators without invariant subsets satisfy the Hypercyclicity Criterion; a competent, modest note with a correct short proof.","tokens_in":8427,"tokens_out":3515,"would_cite":true,"duration_ms":30682,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47A15","47A16"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every operator of Read's type with no non-trivial invariant subset satisfies the Hypercyclicity Criterion.","keywords":["invariant subspace problem","invariant subset problem","operators of Read's type","hypercyclic operators","Hypercyclicity Criterion","direct sum hypercyclicity","topological weak mixing"],"falsifier":"A single counterexample of the type asked in Question 3.1—an operator of Read's type with no non-trivial invariant closed subset whose direct sum $T\\oplus T$ is not hypercyclic—would refute Theorem 3.2. A less decisive but concrete check is to compute $\\|T^{\\lfloor c_n/2\\rfloor}e_0\\|$ for a proposed Read-type construction: if it does not tend to 1 along the sequence selected to approximate $4^k e_0$, then the mechanism of this proof does not apply to that operator.","tokens_in":7599,"feed_emoji":"🔄","tokens_out":6240,"duration_ms":59528,"temperature":0.7,"pith_summary":"The paper proves that if a bounded operator on a separable real or complex Banach space is an operator of Read's type and has no non-trivial invariant closed subset, then its direct sum with itself, $T\\oplus T$, is hypercyclic. Since the Hypercyclicity Criterion is equivalent to $T\\oplus T$ being hypercyclic, this means every known Read-type counterexample to the invariant-subspace problem automatically satisfies the criterion. The proof is short and uses only two structural properties of Read-type operators: the orbit of the first basis vector can approximate any polynomial in the operator at specially chosen times, and the vectors reached halfway through those times stay near norm one. If correct, the result closes off the possibility that Read-type constructions could answer the natural question of whether an operator with no invariant subsets can still fail the Hypercyclicity Criterion.","feed_headline":"Read-type operators pass the Hypercyclicity Criterion","feed_subtitle":"No invariant closed subset means T⊕T is hypercyclic, settling the criterion for all known such operators.","key_machinery":"The carrying object is the sequence of vectors $e_j=T^j e_0$ and the splitting of the special integers $c_n$ into two roughly equal halves inside a lay-off interval: $c_n=\\lfloor c_n/2\\rfloor+(c_n-\\lfloor c_n/2\\rfloor)$. Read-type constructions guarantee two quantitative facts: the approximation property that $\\|T^{c_n}e_0-p(T)e_0\\|$ can be made arbitrarily small for every polynomial $p$ (via a doubling argument from the restricted case $|p|\\le 2$), and the lay-off decay formula that makes $\\|T^{\\lfloor c_n/2\\rfloor}e_0\\|\\to 1$ and the complementary half also tend to norm one. These two facts turn the single hypercyclic vector $e_0$ into a pair of vectors whose joint orbit is dense under $T\\oplus T$.","core_discovery":"The central claim is Theorem 3.2: for any operator $T$ of Read's type acting on a separable Banach space and having no non-trivial invariant subset, $T\\oplus T$ is hypercyclic, equivalently $T$ satisfies the Hypercyclicity Criterion. The proof selects integers $n_k$ such that $T^{c_{n_k}}e_0$ approximates $4^k e_0$, splits each $c_{n_k}$ as $i_{n_k}+j_{n_k}$ with $i_{n_k}=\\lfloor c_{n_k}/2\\rfloor$, and uses the lay-off interval structure to show that $\\|T^{i_{n_k}}e_0\\|\\to 1$ and $\\|T^{j_{n_k}}e_0\\|\\to 1$. Setting $w_k=2^{-k}T^{i_{n_k}}e_0$ and $q_k(T)=2^{-k}T^{j_{n_k}}$ gives $q_k(T)e_0\\to 0$, $w_k\\to 0$, and $q_k(T)w_k\\to e_0$, which by the paper's Proposition 4.2 forces the Hypercyclicity Criterion to hold.","pith_inferences":["A natural extension, not stated in the paper, is to test whether the recent quasinilpotent counterexamples to the invariant-subspace problem share the same two structural properties; if they do, their associated direct sums would also be hypercyclic.","One could conjecture a general principle suggested by the proof: any bounded operator whose nonzero vectors are all hypercyclic and whose construction admits a midpoint splitting with norm-stable intermediate vectors should satisfy the Hypercyclicity Criterion.","The paper's Question 4.4 points toward a sharper dichotomy: measure the size of the hypercyclic vector set for operators that fail the criterion; if those sets are small, the contrast with Read-type operators becomes a quantitative separation between two families of hypercyclic operators."],"forward_implications":["All known operators without non-trivial invariant closed subsets built by Read-type constructions satisfy the Hypercyclicity Criterion, so none can serve as counterexamples to the equivalence between $T\\oplus T$ being hypercyclic and $T$ satisfying the criterion.","The hypercyclic operators discussed in Remark 4.3, which have few non-trivial invariant subsets but do have invariant subspaces, also satisfy the Hypercyclicity Criterion by the same argument.","The proof uses only two structural properties, so the theorem potentially extends to any wider class of operators sharing those two properties, as the paper notes explicitly.","For Read-type operators, having every nonzero vector hypercyclic is strong enough to force the stronger two-dimensional density property of $T\\oplus T$, a regularity phenomenon not shared by all hypercyclic operators."],"supporting_citations":[{"why":"Supplies the general Read-type construction framework, the notation for working and lay-off intervals, and the structural properties used in the proof.","marker":"[16]"},{"why":"Provides the (c)-working intervals and the polynomial approximation property for operators of Read's type, including the doubling argument.","marker":"[15]"},{"why":"Gives the sufficient condition reformulated as Proposition 4.2, which turns the constructed sequences $w_k$ and $q_k(T)$ into the Hypercyclicity Criterion.","marker":"[14]"},{"why":"Establishes the equivalence between $T$ satisfying the Hypercyclicity Criterion and $T\\oplus T$ being hypercyclic, justifying the theorem's phrasing.","marker":"[7]"},{"why":"Provides the contrasting example of a hypercyclic operator whose direct sum is not hypercyclic, motivating Question 3.1 and delimiting the theorem's reach.","marker":"[10]"}],"fun_headline_variants":["T⊕T hypercyclic: Read-type criterion settled","Read-type operators: no invariant set means T⊕T hypercyclic","Read's type: T⊕T hypercyclic when no invariant subsets","Read-type: invariant-free operators get hypercyclic T⊕T"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on two structural guarantees in the construction of operators of Read's type: the orbit of the starting vector can be steered close to any polynomial in the operator at specially chosen times, and the vectors reached halfway through those special times stay close to norm one; if either guarantee fails for some candidate operator, the constructed pair of vectors may not produce a dense orbit for $T\\oplus T$.","fun_headline_variants_meta":{"raw":{"variants":["T⊕T hypercyclic: Read-type criterion settled","Read-type operators: no invariant set means T⊕T hypercyclic","Read's type: T⊕T hypercyclic when no invariant subsets","Read-type: invariant-free operators get hypercyclic T⊕T"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001186,"raw_usage":{"total_tokens":4842,"prompt_tokens":835,"completion_tokens":4007,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":451,"completion_tokens_details":{"reasoning_tokens":3948}},"tokens_in":451,"tokens_out":4007,"duration_ms":27875,"temperature":1.0,"reasoning_tokens":3948,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:36:50.576267+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A single counterexample of the type asked in Question 3.1—an operator of Read's type with no non-trivial invariant closed subset whose direct sum $T\\oplus T$ is not hypercyclic—would refute Theorem 3.2. A less decisive but concrete check is to compute $\\|T^{\\lfloor c_n/2\\rfloor}e_0\\|$ for a proposed Read-type construction: if it does not tend to 1 along the sequence selected to approximate $4^k e_0$, then the mechanism of this proof does not apply to that operator.","supporting_citations":[{"cited_title":"Grivaux and M","cited_arxiv_id":null,"evidence_quote":"Supplies the general Read-type construction framework, the notation for working and lay-off intervals, and the structural properties used in the proof."},{"cited_title":"Grivaux and M","cited_arxiv_id":null,"evidence_quote":"Provides the (c)-working intervals and the polynomial approximation property for operators of Read's type, including the doubling argument."},{"cited_title":"Grivaux, Hypercyclic operators, mixing operators, and the bounded s teps problem, J","cited_arxiv_id":null,"evidence_quote":"Gives the sufficient condition reformulated as Proposition 4.2, which turns the constructed sequences $w_k$ and $q_k(T)$ into the Hypercyclicity Criterion."},{"cited_title":"B` es and A","cited_arxiv_id":null,"evidence_quote":"Establishes the equivalence between $T$ satisfying the Hypercyclicity Criterion and $T\\oplus T$ being hypercyclic, justifying the theorem's phrasing."},{"cited_title":"de la Rosa and C","cited_arxiv_id":null,"evidence_quote":"Provides the contrasting example of a hypercyclic operator whose direct sum is not hypercyclic, motivating Question 3.1 and delimiting the theorem's reach."}],"review_version":1}