REVIEW 2 major objections 5 minor 6 references
The angle along a curve and range-kernel complementarity
T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read A single angle measured along an unbounded curve determines when a bounded operator splits its space into range plus kernel.
desk verdict Genuine new curve-angle characterization with two fixable but load-bearing typos in the printed definitions. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the angle of an operator along an unbounded curve $C$. For vectors $x,y$ with $x\ne 0$, define $s_C(x,y)=\inf_{\lambda\in C}\frac{\|x-\lambda y\|}{\|x\|}$, the relative distance from $x$ to the scaled curve; then $\sin_C A=\inf_{x\notin N(A)} s_C(Ax,x)$, and the intended angle is $\varphi_C(A)=\pi-\arcsin(\sin_C A)$. The load-bearing equivalence is $\varphi_C(A)<\pi$ if and only if there is $c>0$ such that $\|Ax+\lambda x\|\ge c\|Ax\|$ for all $x\in X$ and $\lambda\in C$; this uniform inequality is what Lemma 1 uses to close $R(A)+N(A)$ and separate $R(A)$ from $N(A)$. Proposition 2, an invariant-subspace spectral result, supplies the curve: it keeps the spectrum of $A|_{R(A)}$ away from $D_\infty$.
What would settle it
Read the definition in Section 3.2 literally: with $\varphi_C(A)=\arcsin \sin_C A$, the condition $\varphi_C(A)<\pi$ holds for every operator and every curve, since arcsin never exceeds $\pi/2$. Then any non-complementary operator with closed range and the origin facing the unbounded component, such as a two-dimensional nilpotent Jordan block with the origin on the boundary of the resolvent's unbounded component, satisfies the condition but not the conclusion, so the theorem as printed is false; replacing the definition by $\pi-\arcsin$ and checking whether the proof of Theorem 1 still goes through settles the intended claim.
Extended reading notes
Core claim
Theorem 1 states: for a bounded linear operator $A$ on a Banach space with $0\in D_\infty$ (the origin facing the unbounded component of the resolvent set), $X=R(A)\oplus N(A)$ if and only if $R(A)$ is closed and $\varphi_C(A)<\pi$ for some unbounded curve $C$ emanating from the origin. The forward direction constructs the curve inside the resolvent of the restriction $A|_{R(A)}$, then converts the direct-sum bound $\|Ax+y\|\ge\delta\|Ax\|$ into a uniform angle bound, showing $\varphi_C(A)<\pi$. The reverse direction uses the same angle inequality to prove that $R(A)\cap N(A)=\{0\}$, that $R(A)+N(A)$ is closed, and ultimately that $0$ is not in the spectrum of $A|_{R(A)}$, which forces $R(A^2)=R(A)$ and descent at most one; ascent is already at most one, so complementarity follows. The argument leans on a filling-the-hole result: for every closed invariant subspace $M$, the spectrum of the restriction $A|_M$ avoids $D_\infty$. The definitional caveat is that the manuscript states $\varphi_C(A)=\arcsin \sin_C A$, while every use of the inequality $\varphi_C(A)<\pi$ in the proof requires the motivating definition $\varphi_C(A)=\pi-\arcsin \sin_C A$.
Load-bearing premise
The argument depends on identifying $\varphi_C(A)<\pi$ with the uniform separation inequality $\|Ax+\lambda x\|\ge c\|Ax\|$, which requires the definition $\varphi_C(A)=\pi-\arcsin \sin_C A$; Section 3.2 prints $\varphi_C(A)=\arcsin \sin_C A$, under which $\varphi_C(A)<\pi$ is automatic and Theorem 1 would be vacuous.
Editorial extensions
If this is right
- Range-kernel complementarity becomes equivalent to a quantifiable geometric condition whenever an unbounded curve from the origin avoids the spectrum, replacing chain-length checks with one angle computation.
- The earlier ray-based amplitude-angle theorem is subsumed: a single ray with angle below pi was sufficient but not necessary; with curves, the condition becomes necessary as well, at the cost of allowing curved paths.
- Under complementarity, the proof exhibits a concrete curve (any unbounded path inside the resolvent of $A|_{R(A)}$) along which the angle is strictly below pi.
- If the origin lies in a hole of the spectrum, Proposition 3 implies that every angle along every unbounded curve equals pi despite complementarity, so the geometric criterion distinguishes exactly the case where the origin is not trapped by spectrum.
Reading between the lines
- The constants in the proof connect the margin $\pi-\varphi_C(A)$ to the norm of the projection onto $R(A)$; a quantitative stability statement of that kind is not stated but follows naturally from the argument.
- In Hilbert space, $s_C(Ax,x)$ is a one-parameter infimum, so for polygonal curves the condition $\varphi_C(A)<\pi$ is effectively checkable by quadratic programming; this could yield a computational test for range-kernel complementarity of matrices.
- The same angle construction could be applied to unbounded operators by choosing $C$ inside the resolvent; whether the equivalence survives is a natural open problem the paper does not address.
- The formulas in Sections 3.1 and 3.2 are inconsistent; if the printed arcsin formula were intended, the theorem would be empty, so a corrected manuscript should fix this before the main result is assessed.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces the angle of a bounded linear operator along an unbounded curve emanating from the origin, and uses it to give a spectral characterization of range-kernel complementarity. The main result, Theorem 1, claims that if 0 faces the unbounded component of the resolvent set, then X = R(A) ⊕ N(A) if and only if R(A) is closed and the angle of A along some such curve is less than π. Proposition 3 gives a companion result under the assumption of range-kernel complementarity. The proofs rely on ascent/descent conditions, a closed-range lemma, and the filling-the-hole theorem.
Significance. If the intended statements hold, the paper provides a genuinely geometric criterion for range-kernel complementarity in Banach spaces, extending earlier work on the angle of an operator by replacing rays with arbitrary unbounded curves. The proof strategy is coherent and uses standard tools (ascent/descent equivalence, the filling-the-hole theorem, and approximate point spectrum arguments). The paper is short and the central idea is original relative to the cited literature. However, the printed definition of the curve angle makes the main condition vacuous, and the statement of Theorem 1 has a hypothesis that would trivialize the problem; these issues must be corrected before the claims can be assessed.
major comments (2)
- [Section 3.2, Definition of φ_C(A); Lemma 1; Proposition 3] The printed definition φ_C(A) = arcsin sin_C A makes the condition φ_C(A) < π automatically true for every bounded operator and every curve, because sin_C A ∈ [0,1] implies the principal arcsin lies in [0,π/2]. Under that definition, Lemma 1 would assert that every bounded operator satisfies R(A) ∩ N(A) = {0} and R(A)+N(A) is closed, which is false; for example, the backward shift on ℓ² has R(A)=ℓ² and N(A)=span{e₁}, so R(A)∩N(A)≠{0}. The proof of Lemma 1 actually uses the existence of c>0 such that ‖Ax−λx‖ ≥ c‖Ax‖ for all x∉N(A) and λ∈C, which is equivalent to sin_C A > 0, i.e. to π−arcsin sin_C A < π, not arcsin sin_C A < π. The definition must be corrected to φ_C(A) = π − arcsin sin_C A, matching the discussion in §3.1 and Proposition 1. As printed, Proposition 3 is false and the main theorem is vacuous.
- [Theorem 1 and the note after Proposition 2] The theorem states 'assume 0 ∈ D∞', but D∞ is defined as the unbounded component of the resolvent set, so 0 ∈ D∞ would already imply that A is invertible and range-kernel complementarity is trivial. The abstract's phrase '0 faces the unbounded component' and the remark that the existence of an unbounded curve emanating from the origin is equivalent to 0 ∈ D∞ indicate the intended hypothesis is that 0 belongs to the closure of D∞ and is reachable from D∞ by an unbounded curve avoiding σ(A) except possibly at 0. This distinction is load-bearing: the proof of Theorem 1 uses Proposition 2 to obtain an unbounded curve C ⊆ ρ(A|R(A)) emanating from the origin, which requires 0 to face D∞, not 0 ∈ D∞. Please state the intended hypothesis precisely and adjust all statements (including Proposition 3) accordingly.
minor comments (5)
- [Lemma 1 proof] In the display in the proof of Lemma 1, A(x + y/λ_n) + λ_n(x − y/λ_n) equals Ax + λ_n x − y, not Ax + λ_n x + y; the limit is ‖Ax − y‖. Since −N(A)=N(A), the conclusion is unaffected, but the displayed inequality should be corrected.
- [Theorem 1 converse, paragraph before (7)] The sentence 'By Lemma 1, using the fact that R(A) is closed, we have that R(A)+N(A) is a closed subspace of X which implies that R(A²) is also closed' is not justified as written. Because R(A) and N(A) are closed, intersect trivially, and have closed sum, the associated projection is bounded; combined with (2) this gives a lower bound for ‖Ax‖ in terms of ‖x‖ on R(A), which yields closedness of R(A²). Please add this argument or a reference.
- [Inequality (7)] The inequality (7) states ‖Ax‖ ≥ ‖x‖ for all x∈R(A), but the constant obtained from (2) and the distance to N(A) is generally not 1; either track the constant or normalize the norm in the direct-sum decomposition.
- [Proposition 3 proof] The inequality in the proof of Proposition 3, '‖Axn+λ0xn‖ = ‖Azn+λ0yn+λ0zn‖ ≥ c′‖Azn+λ0zn‖', requires an explicit justification using the bounded projection onto R(A) along N(A) guaranteed by the closed direct sum; the current wording 'since the sum is closed' is too terse.
- [Throughout] There are several typos: 'lenght' in the Introduction, 'Conversly' in the proof of Theorem 1, and 'patricular' in the proof of Proposition 3. Please proofread the manuscript.
Circularity Check
No circularity: the spectral characterization is derived from independent external theorems; the only notable defect is a definitional typo in φ_C, which is a correctness issue, not a circular reduction.
full rationale
The main theorem is not obtained by fitting or by importing a conclusion from the authors' own prior work. Theorem 1 rests on the standard ascent/descent criterion ([4, Prop. 38.4]) and on Radjavi–Rosenthal's corollary of the filling-the-hole theorem ([6, Thm 0.8]); both are external and not self-citations. The authors' earlier paper [1] appears only in the Introduction as background about cosine/amplitude angles and is not used in the proof of Lemma 1 or Theorem 1, so this self-citation is not load-bearing. The new angle φ_C(A) is defined directly from s_C(Ax,x) and sin_C A; it is not defined in terms of range-kernel complementarity, and no parameter is fitted to the conclusion. I flag one non-circular flaw: Section 3.2 prints φ_C(A)=arcsin sin_C A, while §3.1 and Lemma 1 require φ_C(A)=π−arcsin sin_C A. Under the printed definition, sin_C A∈[0,1] makes φ_C(A)<π automatic, so Lemma 1's inference of a uniform c>0 with ‖Ax+λx‖≥c‖Ax‖ is unsupported and the theorem's hypothesis becomes vacuous; the intended corrected definition restores the non-vacuous condition sin_C A>0. Also, '0∈D∞' in Theorem 1 should presumably be '0 faces D∞', since D∞ is the unbounded component of the resolvent set. These are correctness/wording defects, not circular steps, and therefore do not raise the circularity score.
Assumptions & free parameters
assumptions (5)
- standard math Proposition 2 (filling the hole theorem): if M is a closed invariant subspace of A, then σ(A|M)∩D∞=∅.
- standard math X=R(A)⊕N(A) if and only if the ascent and descent of A are at most one.
- standard math ∂σ(A)⊆σapp(A).
- ad hoc to paper The intended definition of φ_C(A) is π − arcsin sin_C A; the printed 'arcsin' is treated as a typo.
- domain assumption The unbounded curve C emanating from 0 can be chosen closed.
Cite this review
Pith. "Pith review of The angle along a curve and range-kernel complementarity." pith.science (2026). https://pith.science/paper/JSVSKYP3
@misc{pith2026190803555,
author = {Pith},
title = {Pith review of: The angle along a curve and range-kernel complementarity},
year = {2026},
howpublished = {\url{https://pith.science/paper/JSVSKYP3}},
note = {Machine review of arXiv:1908.03555}
}
abstract
In this paper, we define the angle of a bounded linear operator $A$ along an unbounded path emanating from the origin and use it to characterize range-kernel complementarity. In particular we show that if $0$ faces the unbounded component of the resolvent set, then $X=R(A)\oplus N(A)$ if and only if $R(A)$ is closed and some angle of $A$ is less than $\pi$.
Reference graph
Works this paper leans on
-
[1]
D. Drivaliaris, N. Yannakakis, The angle of an operator and range - kernel complementarity , J Op Theory 76 (2016), 205–218
work page 2016
-
[2]
Gustafson, The angle of an operator and positive operator products , Bull
K. Gustafson, The angle of an operator and positive operator products , Bull. Amer. Math. Soc. 74 (1968), 488–492
work page 1968
-
[3]
K. Gustafson, D. Rao Numerical range: the field of values of linear operators and m atrices, Springer, New York, (1997)
work page 1997
-
[4]
H. Heuser. Functional Analysis, John Wiley and Sons, (1982)
work page 1982
-
[5]
M. Krein, Angular localization of the spectrum of a multiplicative in tegral in a Hilbert space , Functional Analysis and Its Applications 3 (1969), 73–74
work page 1969
-
[6]
H. Radjavi, P. Rosenthal Invariant subspaces, Springer, New York, (1973). Department of Financial and Management Engineering, Univer sity of the Aegean 41, Kountouriotou Str., 82100 Chios, Greece E-mail address : d.drivaliaris@fme.aegean.gr Department of Mathematics, National Technical University of Athens, Iroon Poly- texneiou 9, 15780 Zografou, Greece E...
work page 1973
Reviewed August 14, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.