REVIEW 2 major objections 7 minor 10 references
General Numerical Radius for Products of Sectorial Matrices
T0 review · 2 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read For sectorial matrices X and Y, every multiplicative unitarily invariant norm N satisfies the product bound ω_N(XY) ≤ sec(α1)sec(α2)ω_N(X)ω_N(Y), and the same inequality holds for the Hadamard product.
desk verdict The main product and Hadamard-product bounds for generalized numerical radius are correct, but Theorem 2.6 is false as stated because the class M^s allows rotations and the proof applies Proposition 2.2 without them. 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 argument rests on three inherited lemmas (Lemmas 2.1–2.3): for a sectorial matrix X, N(X) ≤ sec(α)N(Re X), and two 2×2 block matrices built from Re X, Im X, and X are positive semidefinite. These are combined with the fact that a unitarily invariant norm is Hadamard submultiplicative exactly when it is submultiplicative (Lemma 2.4), and with a diagonal-entry bound for Hadamard products (Lemma 2.5). The generalized numerical radius ω_N(X)=sup_θ N(Re($e^{{iθ}}$X)) is the object being controlled, and the lemmas translate sectoriality into norm bounds that survive multiplication by any multiplicative unitarily invariant N.
What would settle it
Search numerically over pairs of 2×2 sectorial matrices, with α1 and α2 close to π/2, using a multiplicative unitarily invariant norm such as the spectral norm, and check whether ω_N(XY)/(ω_N(X)ω_N(Y)) ever exceeds sec(α1)sec(α2). An affirmative example would refute Theorem 2.1; finding the maximal ratio would also test sharpness.
Extended reading notes
Core claim
The central claim is Theorem 2.1 and Theorem 2.3: whenever X ∈ M^s_{n,α1} and Y ∈ M^s_{n,α2}, the inequalities ω_N(XY) ≤ sec(α1)sec(α2)ω_N(X)ω_N(Y) and ω_N(X∘Y) ≤ sec(α1)sec(α2)ω_N(X)ω_N(Y) hold for every multiplicative unitarily invariant norm N. The proof runs by rotating the matrices into sectors, applying a secant bound to each factor, and, for the Hadamard product, using positive semidefinite block matrices and the Hadamard submultiplicativity of unitarily invariant norms. A companion theorem (Theorem 2.5) refines the Hadamard bound using the diagonal entries of one matrix. Together these results reproduce and extend the known inequalities (6)–(8) for the usual numerical radius ω, which is the special case N = ‖·‖.
Load-bearing premise
The load-bearing premise is that the three inherited sectorial lemmas hold for every multiplicative unitarily invariant norm; the diagonal-entry proof additionally assumes a positive-definite lemma passes to the positive-semidefinite boundary without a written continuity argument.
Editorial extensions
If this is right
- For the classical numerical radius — the case N = operator norm — the theorem reproduces the known bounds ω(XY) ≤ sec(α1)sec(α2)ω(X)ω(Y) and ω(X∘Y) ≤ sec(α1)sec(α2)ω(X)ω(Y).
- For accretive-dissipative matrices (α = π/4), the bounds become ω_N(XY) ≤ 2ω_N(X)ω_N(Y) and ω_N(X∘Y) ≤ 2ω_N(X)ω_N(Y), and for m factors the constant is 2^{m/2}.
- The diagonal refinement (Theorem 2.5) gives ω_N(X∘Y) ≤ sec(α1)sec(α2) max_j |x_jj| ω_N(Y), a sharper Hadamard-product estimate when one factor has small diagonal entries.
- The m-factor versions, Corollaries 2.3 and 2.4, give product-of-secant bounds for products of any finite number of sectorial matrices.
Reading between the lines
- The proof mechanism suggests the secant factor may be improvable for specific norms other than the operator norm; checking extremal examples for Schatten p-norms would be a natural test of sharpness.
- Because the argument never uses dimension in an essential way, a Hilbert-space operator analogue of these bounds likely holds for suitable classes of unitarily invariant norms, though the cited sectorial lemmas would need to be verified in that setting.
- The diagonal-entry refinement could be practically useful for bounding the spectral radius of Hadamard products of structured matrices, since it only requires diagonal data of one factor.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the generalized numerical radius ω_N(X)=sup_θ N(Re(e^{iθ}X)) for a multiplicative unitarily invariant self-adjoint norm N, and applies it to matrices whose numerical ranges are contained in sectors of the complex plane up to a unimodular rotation (the class M^s_{n,α}). The main results are Theorem 2.1, giving ω_N(XY) ≤ sec(α_1)sec(α_2)ω_N(X)ω_N(Y), and Theorem 2.3, giving the analogous bound for the Hadamard product ω_N(X∘Y). Further results give bounds using diagonal entries (Theorem 2.5) and a bound involving Re X and Re Y (Theorem 2.6). Several corollaries specialize to known inequalities for the usual numerical radius.
Significance. If the central product inequalities hold, they give a uniform generalization of the known numerical-radius bounds (6) and (7) to arbitrary multiplicative unitarily invariant norms, with constants independent of the dimension and of the chosen norm. The proofs are mostly transparent and rely on published lemmas with independent proofs; there are no fitted parameters or circular normalizations. The main weakness is that Theorem 2.6 is false as stated, so the manuscript cannot be accepted in its current form. The defect is localized to Section 2.3 and does not appear to infect Theorems 2.1 and 2.3.
major comments (2)
- [§2.3, Theorem 2.6] Theorem 2.6 is false as stated. The proof applies Proposition 2.2 to X, but Proposition 2.2 requires W(X)⊂S_{α_1}, whereas X∈M^s_{n,α_1} only guarantees a unit z with W(zX)⊂S_{α_1}. For a concrete counterexample, take n=1, X=iI, Y=I, N the spectral norm, and α_1=π/4. Since W(-iX)=W(I)={1}⊂S_{π/4}, we have X∈M^s_{1,π/4}; however ω_N(X∘Y)=ω_N(iI)=1, while (1+tan(π/4))ω_N(Re X)ω_N(Y)=2·0·1=0. The theorem should either assume W(X)⊂S_{α_1} and W(Y)⊂S_{α_2} directly, or replace Re X and Re Y by Re(zX) and Re(wY) with the rotations made explicit. The derivation of the displayed consequence (19) is therefore also invalid as written, although (19) may be recoverable by first rotating X and Y before applying Proposition 2.2.
- [§2.3, proof of Theorem 2.5] The proof of Theorem 2.5 skips two justifications that should be stated. First, the passage from inequality (15), which involves N, to the displayed bound involving ω_N(Re(zX)∘Re(wY)) is valid only because Re(zX)∘Re(wY) is Hermitian and for Hermitian H one has ω_N(H)=N(H). Second, Lemma 2.6 is applied with the positive definite matrix Re(zX), and this positivity should be stated explicitly; it holds because W(zX)⊂S_{α_1} implies Re(zX)>0. These are local fixable gaps rather than errors, but the proof should not leave them implicit.
minor comments (7)
- [Theorem 2.1] The statement says 'with α_2, α_2 ∈ [0,π/2)' but should say α_1, α_2.
- [Theorem 2.6 consequence] The line 'where α = max{α_1, α_1}' should read α = max{α_1, α_2}.
- [Corollary 2.5] The product term 'X1X1...Xm' and the repeated 'ω_N(Xm)' appear to be typos; the intended statement is a product of distinct matrices X_1,...,X_m.
- [Theorem 2.2] The phrase 'such that such that at least one' contains a duplicated 'such that'.
- [Lemma 2.6] The notation 'w_N' is used in the statement instead of 'ω_N'; this should be harmonized.
- [Corollary 2.10] The formula contains a misplaced parenthesis: 'min {(' should be 'min {'.
- [Theorem 2.3, proof] The invariance ω_N(X∘Y)=ω_N((zX)∘(wY)) for |z|=|w|=1 is used implicitly and should be stated explicitly.
Circularity Check
No circularity; the proofs reduce to independent published sectorial-matrix lemmas and standard norm inequalities, with no fitted input or definitional self-reference.
full rationale
The derivation chain is not circular. Theorem 2.1 bounds omega_N(XY) by first using (11), multiplicativity of N, and Lemma 2.1 on the rotated matrices zX and wY; Lemma 2.1 bounds N(zX) in terms of N(Re zX), not in terms of the product inequality being proved. Theorem 2.3 similarly uses Lemma 2.3 blocks and Hadamard submultiplicativity, and Theorem 2.5 uses Lemma 2.6 and diagonal entries; none of these steps reintroduces the target bound as an assumption. The Lemmas 2.1-2.3 are cited from the author's earlier papers [2]-[4], but they are published results with stated hypotheses (W(X) subset S_alpha) that do not contain the present omega_N product claims, so under the review rule they count as independent evidence rather than circular self-support. No parameter is fitted and no prediction is equivalent by construction to an input. A separate correctness defect is worth noting but is not circularity: the proof of Theorem 2.6 applies Proposition 2.2 to an unrotated X in M^s_{n,alpha}, for which W(X) subset S_alpha need not hold; e.g., X=iI, Y=I with the spectral norm gives omega_N(X circ Y)=1 but the asserted bound is 0. This invalidates Theorem 2.6 as stated, but it is a rotational-application gap, not a circular reduction.
Assumptions & free parameters
assumptions (6)
- domain assumption For sectorial X (W(X) subset S_alpha), N(X) <= sec(alpha) N(Re(X)) for any unitarily invariant norm N.
- domain assumption For sectorial X, the block matrix [[tan(alpha) Re(X), Im(X)], [Im(X), tan(alpha) Re(X)]] is positive semidefinite.
- domain assumption For sectorial X, the block matrix [[sec(alpha) Re(X), X], [X*, sec(alpha) Re(X)]] is positive semidefinite.
- standard math A unitarily invariant norm is Hadamard submultiplicative if and only if it is submultiplicative (Lemma 2.4, Horn-Johnson).
- standard math For Y > 0, N(X o Y) <= max_i y_ii N(X) (Lemma 2.5, Horn-Johnson).
- domain assumption N is multiplicative, unitarily invariant, and self-adjoint.
Cite this review
Pith. "Pith review of General Numerical Radius for Products of Sectorial Matrices." pith.science (2026). https://pith.science/paper/EWFD5OQP
@misc{pith2026250602042,
author = {Pith},
title = {Pith review of: General Numerical Radius for Products of Sectorial Matrices},
year = {2026},
howpublished = {\url{https://pith.science/paper/EWFD5OQP}},
note = {Machine review of arXiv:2506.02042}
}
abstract
In this paper, we investigate the generalized numerical radius $\omega_N$, associated with a matrix norm $N$ defined by $\omega_N(X) = \sup_{\theta \in \mathbb{R}} N(\operatorname{Re}(e^{i\theta}X))$. We focus on matrices whose numerical ranges are contained in sectors of the complex plane (sectorial matrices) and derive upper bounds for $\omega_N(XY)$ and $\omega_N(X \circ Y)$ for such matrices $X$ and $Y$. Our results generalize and refine well known numerical radius inequalities. Several known inequalities for $\omega(X)$ are recovered as special cases.
Reference graph
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Reviewed August 7, 2026 · model on record in the stance chip above.
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