{"id":"f0da522b-93b0-4a17-af2f-d1a36b7adde4","arxiv_id":"1908.04105","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors prove that bounded components of the quasi-Fredholm and semi B-Fredholm resolvent sets correspond one-to-one, and use this to characterize when compact perturbations preserve SVEP.","lead":"This paper analyzes the quasi-Fredholm spectrum, a refined notion of an operator's spectrum, and how it behaves under compact perturbations. It gives conditions under which the single-valued extension property survives such perturbations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main equivalence in Theorem 3.6 rests on an unverified external implication from Zeng–Zhong–Jiang [9], and the paper never states the hypotheses under which connectedness of ρ_qf(T) forces ρ_qf(T)=ρ(T)∪Π(T).","rationale":"The reader's weakest-assumption analysis correctly identifies the external theorem from [9] as the load-bearing point: without it, Lemma 2.11 fails to establish ρ_qf(T) = ρ(T) ∪ Π(T), and Theorem 3.1, Theorem 3.5, and Theorem 3.6 lose their main engine. I agree with this diagnosis. The secondary point in Theorem 2.2 is a gap in exposition rather than the central issue: the claim that Ω ⊂ σ_sf(T) implies Ω ⊂ iso σ_sf(T) can be repaired by noting that an open subset of σ_sf cannot consist of isolated points, and the countable-removal step in the converse direction is standard in the plane. The main concern is therefore not an internal inconsistency but an under-specified reliance on an imported result. Since the reader already marked the paper CONDITIONAL for exactly this reason, my stress-test does not change the verdict. If checking [9] reveals that the missing hypotheses are present and the implication holds with no extra SVEP assumption, the paper can likely be upgraded to ACCEPT; if the citation is misapplied, the central theorems would need revision.","tokens_in":10495,"tokens_out":12649,"duration_ms":121921,"concrete_test":"Obtain the full text of [9] (Zeng, Zhong, Jiang, Glas. Mat. Ser. III 50(70) (2015), 429–440) and transcribe the exact statements and hypotheses of Theorems 3.6 and 3.7. Then check two points: (a) Does connectedness of ρ_qf(T), together with ρ(T) ≠ ∅, alone imply ρ_qf(T) = ρ(T) ∪ Π(T), or is an explicit SVEP assumption needed? (b) Does Theorem 3.5's use of [9] require an additional hypothesis that is not stated in the present paper? If either cited theorem needs extra hypotheses, identify where that hypothesis is absent in Lemma 2.11, Theorem 3.1, and Theorem 3.6, and recompute those proofs with the added assumption to see whether the main characterization still holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central permanence theorem (Theorem 3.6) and its supporting results Theorem 3.1, Theorem 3.5, and Theorem 3.8 all depend on one imported fact: from connectedness of ρ_qf(T) (together with ρ(T) ⊂ ρ_qf(T)) one may conclude that ρ_qf(T) = ρ(T) ∪ Π(T), equivalently that p(λI − T) = q(λI − T) < ∞ for every λ ∈ ρ_qf(T). The paper invokes [9, Theorems 3.6 and 3.7] for this in Lemma 2.11 and again in Theorem 3.1, but it never states the hypotheses, exact statements, or proof context of those theorems. In particular, the direction used in Lemma 2.11 and Theorem 3.1 — connectedness alone implying every point of ρ_qf is a pole of the resolvent — is a very strong assertion. The same pair of cited theorems is also used in two different directions inside Theorem 3.5: once to convert connectedness into SVEP on ρ_qf, and once to convert SVEP on a component Ω into the inclusion Ω ⊂ ρ_a ∪ iso σ_a. If [9] requires an additional SVEP hypothesis, or requires the component to intersect ρ_a in a way that is not automatic in Lemma 2.11, then the proofs of Theorem 3.1, Theorem 3.5, and the main Theorem 3.6 do not go through as written. I am not claiming the result is false; I am identifying a load-bearing step that the manuscript leaves unverified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the quasi-Fredholm resolvent set ρ_qf(T)=C\\σ_qf(T) for bounded operators on Banach and Hilbert spaces. Section 2 establishes relations between the components of ρ_sf(T), ρ_sbf(T), ρ_qf(T), and ρ_Γ(T): Theorem 2.2 asserts that connectedness of ρ_sbf(T) and ρ_sf(T) are equivalent; Theorems 2.4–2.9 relate bounded components of the four resolvent sets and prove a one-to-one correspondence under empty-interior assumptions; Theorem 2.13 gives a four-way equivalence of connectedness plus empty interior for the four spectra. Section 3 applies these results to compact perturbations: Theorem 3.1 gives a decomposition of σ(T+K) when ρ_qf(T) is connected, Theorem 3.5 characterizes when T+K has SVEP on ρ_qf(T+K) for every compact K, Theorem 3.6 gives an equivalence between stability of SVEP under all compact perturbations and connectedness plus empty interior of the relevant resolvent spectra, and Theorem 3.8 treats the case σ_qf(T)=∅.","tokens_in":10789,"tokens_out":10808,"duration_ms":108915,"significance":"If the main results are valid, the paper gives a clean spectral characterization of when SVEP is permanent under compact perturbations, extending and unifying work of Shi [8] and Zhu–Li [10]. The component-correspondence results in Section 2 are concrete and testable, and the paper is organized around a natural and worthwhile question. A visible strength is that the arguments are mostly built from standard Fredholm theory rather than from ad hoc constructions. The significance is, however, conditional: the central equivalence in Section 3 depends on an imported assertion from [9] whose hypotheses are never stated or verified in the manuscript, so the main theorem cannot currently be checked by a reader.","major_comments":[{"comment":"The load-bearing step is the assertion that connectedness of ρ_qf(T) implies ρ_qf(T)=ρ(T)∪Π(T). In Lemma 2.11 the text reads 'As ρ_qf(T) is connected and ρ(T)⊂ρ_qf(T), by [9, Theorems 3.6, 3.7] p(λI−T)=q(λI−T)<∞ for all λ∈ρ_qf(T)', and Theorem 3.1 uses the same implication for T and for T+K. The manuscript never states the exact content of [9, Theorems 3.6 and 3.7], nor verifies that their hypotheses are satisfied. If those theorems require, for instance, SVEP on the component in question, the arguments in Lemma 2.11 and Theorem 3.1 have no such hypothesis available. This gap must be repaired by quoting the theorems and providing a verification, or by adding the missing hypotheses to the statements of Lemma 2.11, Theorem 3.1, Theorem 3.5, and Theorem 3.6.","section":"Lemma 2.11; Theorem 3.1"},{"comment":"The sentence 'If Ω∩ρ_sf(T)=∅, then Ω⊂σ_sf(T) which implies that Ω⊂iso σ_sf(T)' is not justified as written: an open connected subset of σ_sf(T) need not consist of isolated points. The intended inference is that Ω⊂ρ_sbf(T), and since σ_sf(T)=σ_sbf(T)∪iso σ_sf(T), the intersection σ_sf(T)∩ρ_sbf(T) is exactly iso σ_sf(T). This additional argument should be stated explicitly. In the first paragraph of the same proof, the removal of the at-most-countable set iso σ_sf(T) from a connected open set is asserted to preserve connectedness; this is plausible only because iso σ_sf(T) is discrete inside ρ_sbf(T), and the authors should justify it.","section":"Theorem 2.2, second paragraph"},{"comment":"The proof applies Theorem 2.16 to conclude that ρ_qf(T+K) is connected, but Theorem 2.16 has as an explicit hypothesis int σ_p(T)=∅. No argument is given that σ_qf(T)=∅ implies int σ_p(T)=∅, and the line 'As int σ_qf(T)=∅' supplies only a different condition that is automatically true when σ_qf(T)=∅. This step needs either a proof of the missing implication or a different argument for connectedness of ρ_qf(T+K).","section":"Theorem 3.8"}],"minor_comments":[{"comment":"The statement says 'for any compact operator K∈K(X)' in a Hilbert-space result; it should be K(H).","section":"Theorem 3.8"},{"comment":"The conclusion 'if ρ_qf(T) consists of finite bounded components, then ρ_Γ(T) consists of bounded components' should read 'consists of finitely many bounded components', and the claim that distinct bounded components of ρ_Γ(T) yield distinct components of ρ_qf(T) is asserted in one sentence and needs a short proof.","section":"Remark 2.15"},{"comment":"There are several typographical errors, including 'opeartor', 'F redholm' in the abstract, 'if is a lower semi B-Fredholm', 'rhosf' in the proof of Theorem 2.13, and 'σusbb' in Example 3.4, which should be corrected.","section":"Throughout"},{"comment":"The switch from 'T+K has SVEP at every point of ρ_qf(T+K) for any K' to 'T+K+K1 has SVEP at every point of ρ_qf(T+K+K1)' should be made explicit: the latter follows only because K+K1 is again an arbitrary compact perturbation, and the text should say so.","section":"Theorem 3.5"},{"comment":"The proof is described as 'following the lines of the proof of Theorem 2.2'; since Theorem 2.2 needs the repair described in the major comments, the same repair should be applied to the two parts of Theorem 2.3.","section":"Theorem 2.3"}],"recommendation":"major_revision","confidential_remarks":"The main risk is that the unstated theorem from [9] may require an SVEP hypothesis on the component. If so, the forward direction of Theorem 3.5 already has such a hypothesis, but Lemma 2.11 and Theorem 3.1 do not, and the main equivalence in Theorem 3.6 would need reformulation. I recommend asking the authors to state and verify the quoted results from [9] before the paper can be assessed further."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the quick take: the paper is a workmanlike extension of known component theorems to the quasi-Fredholm spectrum, and Section 2 stands up. The problem is Section 3, which repeatedly leans on an imported result from Zeng–Zhong–Jiang [9] that is never stated or verified.\n\nWhat is actually new: The component correspondence between ρ_sbf(T) and ρ_qf(T) (Theorems 2.4, 2.7) and the equivalence of connectedness of the four resolvent sets ρ_sf, ρ_sbf, ρ_qf, ρ_Γ (Theorem 2.13) are new statements, and the proof technique is competent. The bookkeeping in Theorem 2.2 is compressed but correct: the step \"Ω⊂σ_sf(T) implies Ω⊂iso σ_sf(T)\" is fine because Ω⊂ρ_sbf(T), so Ω avoids σ_sbf(T), and Lemma 2.1 gives σ_sf = σ_sbf ∪ iso σ_sf. The paper is honestly positioned as an adaptation of Shi [8] and Zeng et al. [9], and there is no circularity or parameter fitting.\n\nThe soft spot is real and load-bearing. Lemma 2.11 and Theorem 3.1 assert that if ρ_qf(T) is connected, then every λ∈ρ_qf(T) has p(λI-T)=q(λI-T)<∞, so ρ_qf(T)=ρ(T)∪Π(T), citing [9, Theorems 3.6, 3.7]. The paper never states those theorems or their hypotheses. This is a strong claim: connectedness of a resolvent-type set alone (plus the always-true inclusion ρ(T)⊂ρ_qf(T)) is being used to force local SVEP and poles. Maybe [9] proves exactly this—if a component of ρ_qf intersects ρ(T), then it consists of resolvent points and poles—and if so the paper goes through. But the authors owe the reader the exact statements and a check that the hypotheses hold. The same imported fact is used in both directions of Theorem 3.5, so Theorem 3.6 inherits the vulnerability. If the import turns out to require extra assumptions, Theorem 3.1, Theorem 3.5, and Theorem 3.6 collapse.\n\nMinor issues: several typos (\"rhosf\", \"opeartor\"), Example 3.4 has a notation slip, and Theorem 3.8's proof chain depends on Theorem 2.16, which itself uses Remark 2.15—also fine but thin.\n\nWho this is for: people working in Fredholm and local spectral theory. Section 2 alone is a modest but defendable contribution. Section 3 is conditional on [9] being exactly as needed. I'd send this to a referee; the referee can force the authors to spell out [9] and either the paper survives or gets substantially revised. Not a desk reject.","headline":"Solid component theory for the quasi-Fredholm resolvent, but the permanence theorem rests on an unstated import from [9] and needs a major revision.","tokens_in":11387,"tokens_out":8176,"would_cite":true,"duration_ms":80271,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47A10","47A55","47B15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that SVEP survives every compact perturbation of a Hilbert-space operator exactly when the quasi-Fredholm resolvent is connected and the quasi-Fredholm spectrum has empty interior.","keywords":["quasi-Fredholm spectrum","quasi-Fredholm resolvent","topological uniform descent","SVEP","compact perturbation","semi B-Fredholm spectrum"],"falsifier":"A decisive check would be to exhibit a bounded Hilbert-space operator $T$ with $\\rho_{qf}(T)$ connected and $\\operatorname{int}\\sigma_{qf}(T)=\\emptyset$ but some compact $K$ making $T+K$ fail SVEP, for instance by having a non-isolated point in its point spectrum; Theorem 3.6 says no such pair exists. A less computational check is to test the step in Theorem 2.2 that a bounded component of $\\rho_{sbf}(T)$ contained in $\\sigma_{sf}(T)$ is isolated in $\\sigma_{sf}(T)$, since Theorem 2.13 and hence Theorem 3.6 depend on that inference.","tokens_in":10219,"feed_emoji":"","tokens_out":10395,"duration_ms":96072,"temperature":0.7,"pith_summary":"This paper studies the quasi-Fredholm resolvent set $\\rho_{qf}(T)$, the points where $\\lambda I - T$ is quasi-Fredholm, and uses it to decide when the single-valued extension property (SVEP) survives compact perturbations. The main Hilbert-space theorem states that for a bounded operator $T$, the operator $T+K$ has SVEP for every compact $K$ if and only if $\\rho_{qf}(T)$ is connected and the quasi-Fredholm spectrum $\\sigma_{qf}(T)$ has empty interior; the same condition is shown equivalent to analogous connectedness-plus-empty-interior conditions for the semi-Fredholm, semi-B-Fredholm, and topological-uniform-descent resolvent sets. On Banach spaces, the paper establishes a structural result: when the semi-B-Fredholm spectrum has empty interior, the bounded components of $\\rho_{sbf}(T)$ and $\\rho_{qf}(T)$ correspond one-to-one. A sympathetic reader would care because it gives a spectral-shape criterion, with no extra parameters, for a robust local spectral property under the broadest class of compact perturbations.","feed_headline":"Quasi-Fredholm spectrum fixes when SVEP survives compact perturbations","feed_subtitle":"The criterion: connected quasi-Fredholm resolvent, empty-interior spectrum, stable under every compact K.","key_machinery":"The load-bearing object is the quasi-Fredholm resolvent set $\\rho_{qf}(T)=\\mathbb{C}\\setminus\\sigma_{qf}(T)$, where $\\lambda I-T$ is quasi-Fredholm when its iterated ranges are closed from some point on and satisfy a finite stable-descent condition; the paper also uses the larger semi-B-Fredholm resolvent $\\rho_{sbf}(T)$ and the topological-uniform-descent resolvent $\\rho_{\\Gamma}(T)$, which sit between $\\rho_{qf}(T)$ and the semi-Fredholm resolvent $\\rho_{sf}(T)$. The identity doing the work, imported as a cited theorem, is that connectedness of $\\rho_{qf}(T)$ forces $\\rho_{qf}(T)=\\rho(T)\\cup\\Pi(T)$, with $\\Pi(T)$ the poles of the resolvent; this converts a question about connected components into a question about where the operator is invertible, and that conversion is what lets compact perturbations, which preserve semi-Fredholm spectra, be controlled.","core_discovery":"The central claim is Theorem 3.6: for $T \\in B(H)$ on a Hilbert space, the following are equivalent: $T+K$ has SVEP for every compact operator $K$; the adjoint $T^*+K$ has SVEP for every compact $K$; and each of the four resolvent sets $\\rho_{sf}(T)$, $\\rho_{sbf}(T)$, $\\rho_{qf}(T)$, and $\\rho_{\\Gamma}(T)$ is connected with the corresponding spectrum having empty interior; moreover the same four conditions hold after any compact perturbation. The paper argues that connectedness of the quasi-Fredholm resolvent is the organizing fact: it forces $\\rho_{qf}(T)=\\rho(T)\\cup\\Pi(T)$, so the quasi-Fredholm resolvent differs from the ordinary resolvent only by poles, and this equality is preserved under compact perturbations in a way that makes SVEP stable. If the theorem is right, the behavior of SVEP under compact perturbations is completely controlled by a short list of spectral shape conditions.","pith_inferences":["A natural testable extension is whether the same four-way equivalence survives for Riesz perturbations or for operators on Banach spaces with a localized SVEP assumption; the paper itself states the characterization only for compact perturbations on Hilbert space.","The unilateral-shift example shows that SVEP on each point of $\\rho_{qf}$ does not force connectedness, so the equivalence in Theorem 3.5 is not a tautology; relaxing the compact perturbations to, say, perturbations small in norm may still be governed by the same resolvent shape.","A careful reader should audit the step in Theorem 2.2 where a bounded component of $\\rho_{sbf}(T)$ contained in $\\sigma_{sf}(T)$ is concluded to be isolated in $\\sigma_{sf}(T)$; if that step needs repair, the exact statement of Theorem 2.13 and hence the 'if and only if' in Theorem 3.6 may narrow.","Because compact perturbations preserve semi-Fredholm spectra, the result suggests that the quasi-Fredholm resolvent is the right object through which to study stability of local spectral properties under the ideal of compact operators; a computational check on concrete operators such as weighted shifts or diagonal-plus-rank-one operators would test the sharpness of the empty-interior condition."],"forward_implications":["For Hilbert-space operators, SVEP is stable under every compact perturbation exactly when any one, hence all, of the four resolvent sets is connected and the corresponding spectrum has empty interior (Theorem 3.6).","If a compact perturbation destroys SVEP, then the quasi-Fredholm resolvent is disconnected or its spectrum has interior; no finer invariant is needed.","When $\\rho_{qf}(T)$ is connected, the spectrum of $T+K$ splits as $\\sigma_{qf}(T+K)\\cup\\Pi(T+K)\\cup(\\sigma_{sbf}(T+K)\\setminus\\sigma_{qf}(T+K))$ for every compact $K$ (Theorem 3.1).","If $\\sigma_{qf}(T)=\\emptyset$, then every compact perturbation has spectrum consisting only of isolated semi-B-Fredholm points plus poles (Theorem 3.8).","On Banach spaces, if the semi-B-Fredholm spectrum has empty interior, the bounded components of $\\rho_{sbf}(T)$ and $\\rho_{qf}(T)$ are in bijection (Theorem 2.7)."],"supporting_citations":[{"why":"Supplies the theorem that connectedness of $\\rho_{qf}(T)$ forces SVEP on that set and $\\rho_{qf}(T)=\\rho(T)\\cup\\Pi(T)$; this is the main engine of the paper's equivalences.","marker":"[9]"},{"why":"Supplies the topological-uniform-descent results, including $\\rho_{\\Gamma}(T)=\\rho_{sf}(T)\\cup E$ with $E\\subset\\operatorname{iso}\\sigma_{sf}(T)$, and the compact-perturbation corollaries used in Theorems 2.4, 3.5, and 3.6.","marker":"[8]"},{"why":"Provides the spectral-inclusion framework $\\sigma_{\\Gamma}(T)\\subset\\sigma_{qf}(T)\\subset\\sigma_{sbf}(T)\\subset\\sigma_{sf}(T)$ and the localization facts about semi-B-Fredholm operators underlying many proofs.","marker":"[2]"},{"why":"Supplies the compact-perturbation characterizations of SVEP in terms of positivity of index on semi-Fredholm resolvent components, used directly in Theorems 3.3 and 3.7.","marker":"[10]"},{"why":"Supplies the local spectral theory result used in Theorem 3.5 to pass SVEP from the perturbed operator to the semi-Fredholm resolvent.","marker":"[1]"},{"why":"Provides the computation of the quasi-Fredholm spectrum of the unilateral shift used in Example 3.4 to show that SVEP on $\\rho_{qf}$ does not imply connectedness.","marker":"[11]"},{"why":"Supplies the decomposition $\\sigma_{uw}=\\sigma_{usbw}\\cup\\operatorname{iso}\\sigma_{uw}$ and its Weyl analogue, used in Theorem 2.3.","marker":"[5]"}],"fun_headline_variants":["SVEP survives compact perturbations iff quasi-Fredholm resolvent is connected","Connected quasi-Fredholm resolvent ensures SVEP under compact perturbations","Theorem 3.6: SVEP stable under compact K when spectra have empty interior","Quasi-Fredholm connectedness: the key to SVEP persistence under compact K"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on a cited result that connectedness of the quasi-Fredholm resolvent already gives SVEP there and forces $\\rho_{qf}(T)=\\rho(T)\\cup\\Pi(T)$; the paper invokes that result without verifying its hypotheses, and if it needs extra conditions the main equivalences no longer follow.","fun_headline_variants_meta":{"raw":{"variants":["SVEP survives compact perturbations iff quasi-Fredholm resolvent is connected","Connected quasi-Fredholm resolvent ensures SVEP under compact perturbations","Theorem 3.6: SVEP stable under compact K when spectra have empty interior","Quasi-Fredholm connectedness: the key to SVEP persistence under compact K"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000831,"raw_usage":{"total_tokens":3562,"prompt_tokens":814,"completion_tokens":2748,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":430,"completion_tokens_details":{"reasoning_tokens":2664}},"tokens_in":430,"tokens_out":2748,"duration_ms":20141,"temperature":1.0,"reasoning_tokens":2664,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:54:21.366800+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive check would be to exhibit a bounded Hilbert-space operator $T$ with $\\rho_{qf}(T)$ connected and $\\operatorname{int}\\sigma_{qf}(T)=\\emptyset$ but some compact $K$ making $T+K$ fail SVEP, for instance by having a non-isolated point in its point spectrum; Theorem 3.6 says no such pair exists. A less computational check is to test the step in Theorem 2.2 that a bounded component of $\\rho_{sbf}(T)$ contained in $\\sigma_{sf}(T)$ is isolated in $\\sigma_{sf}(T)$, since Theorem 2.13 and hence Theorem 3.6 depend on that inference.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that connectedness of $\\rho_{qf}(T)$ forces SVEP on that set and $\\rho_{qf}(T)=\\rho(T)\\cup\\Pi(T)$; this is the main engine of the paper's equivalences."},{"cited_title":"Shi, Topological uniform descent and compact perturbations , Rev","cited_arxiv_id":null,"evidence_quote":"Supplies the topological-uniform-descent results, including $\\rho_{\\Gamma}(T)=\\rho_{sf}(T)\\cup E$ with $E\\subset\\operatorname{iso}\\sigma_{sf}(T)$, and the compact-perturbation corollaries used in Theorems 2.4, 3.5, and 3.6."},{"cited_title":"Aiena, Fredholm and local spectral theory II , Lecture Notes in Mathematics, 2235, Springer, Cham, 2018","cited_arxiv_id":null,"evidence_quote":"Provides the spectral-inclusion framework $\\sigma_{\\Gamma}(T)\\subset\\sigma_{qf}(T)\\subset\\sigma_{sbf}(T)\\subset\\sigma_{sf}(T)$ and the localization facts about semi-B-Fredholm operators underlying many proofs."},{"cited_title":"Zhu and C","cited_arxiv_id":null,"evidence_quote":"Supplies the compact-perturbation characterizations of SVEP in terms of positivity of index on semi-Fredholm resolvent components, used directly in Theorems 3.3 and 3.7."},{"cited_title":"Aiena, Fredholm and local spectral theory, with applications to mu ltipliers, Kluwer Academic Publishers, Dordrecht, 2004","cited_arxiv_id":null,"evidence_quote":"Supplies the local spectral theory result used in Theorem 3.5 to pass SVEP from the perturbed operator to the semi-Fredholm resolvent."},{"cited_title":"ˇZivkovi´ c-Zlatanovi´ c and M","cited_arxiv_id":null,"evidence_quote":"Provides the computation of the quasi-Fredholm spectrum of the unilateral shift used in Example 3.4 to show that SVEP on $\\rho_{qf}$ does not imply connectedness."},{"cited_title":"Berkani and H","cited_arxiv_id":null,"evidence_quote":"Supplies the decomposition $\\sigma_{uw}=\\sigma_{usbw}\\cup\\operatorname{iso}\\sigma_{uw}$ and its Weyl analogue, used in Theorem 2.3."}],"review_version":1}