REVIEW 5 minor 8 references
Normal operators with highly incompatible off-diagonal corners
T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read For every finite dimension $n\ge2$ and every pair of ranks $1\le j,k\le\lfloor n/2\rfloor$, some normal matrix and projection realize corner ranks $k$ and $j$; in infinite dimension, every pair of ranks from $0$ to $\infty$ occurs.
desk verdict A solid, honest paper that closes the finite-dimensional rank-pair question with explicit constructions; worth serious refereeing despite some compressed presentation. 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 load-bearing device is a Schur-product rank-transfer construction. Given diagonal normal blocks $A=\mathrm{diag}(\alpha_1,\dots,\alpha_m)$ and $B=A^*$, the difference matrix $Z_{jk}=\alpha_j-\beta_k$ records where cancellation can happen; the paper chooses a positive definite matrix $S$ so that the entrywise product $S\bullet Z$ has rank one while the transposed product $S^t\bullet Z$ has full rank $m$. Factoring $S=MN^{-1}$ with commuting positive contractions $M,N$ satisfying $M^2+N^2=I$ produces the projection $P=\begin{bmatrix}M^2&MN\\MN&N^2\end{bmatrix}$, and after conjugation by isometries the corner ranks become exactly the ranks of those two Schur products. A Toeplitz-matrix invertibility estimate supplies the required $S$ by taking the parameter $\gamma$ large. The same block-projection formula, with diagonal weights $\alpha_n=1/\sqrt{1+4^{-n}}$ chosen so that cross terms cancel exactly, is used in infinite dimension to make one corner zero and the other a quasiaffinity.
What would settle it
Using the paper's explicit data for $m=3$ (so $n=6$), take $A=\mathrm{diag}(\gamma+i,2\gamma+i,3\gamma+i)$, $B=A^*$, $\gamma=100$, $S_{jk}=2i/((j-k)\gamma+2i)$, and build $P$ from $S=MN^{-1}$; the theorem predicts $\operatorname{rank}D_2=3$ and $\operatorname{rank}D_3=1$ for $D=\mathrm{diag}(A,B)$, so any deviation is a direct counterexample. Alternatively, exhibiting a normal operator $D$ on $H\oplus H$ with $D_2$ invertible and $D_3$ compact would refute Proposition 3.3.
Extended reading notes
Core claim
The central discovery is a complete realization theorem for the rank pair $(\operatorname{rank}D_2,\operatorname{rank}D_3)$. Theorem 2.5 states that for every $n\ge2$ and every $1\le j,k\le\lfloor n/2\rfloor$ there exists a normal $D\in M_n(\mathbb{C})$ and an orthogonal projection $P$ with $\operatorname{rank}D_2=k$ and $\operatorname{rank}D_3=j$, and this is best possible because each corner acts between subspaces of dimension at most $\lfloor n/2\rfloor$. Theorem 3.2 extends the statement to an infinite-dimensional separable Hilbert space: for all $0\le j,k\le\infty$ such a normal operator and projection exist. The paper also establishes a sharp infinite-dimensional asymmetry: a normal operator can have a zero lower-left corner while the upper-right corner is a quasiaffinity, but it cannot have an invertible upper-right corner and a compact lower-left corner.
Load-bearing premise
The construction leans on two borrowed results it does not re-prove: the low-dimensional classification of property (CR) used as the base of the induction, and a theorem about self-commutators in infinite dimension; if either is wrong, the corresponding construction collapses.
Editorial extensions
If this is right
- In any finite-dimensional Hilbert space, the attainable rank pairs for corners of normal operators are exactly the rectangle $\{1,\dots,\lfloor n/2\rfloor\}^2$; no hidden spectral constraint remains.
- The finite-dimensional rank-compatibility question is closed: property (CR) describes the case where the rectangle collapses to the diagonal, while the new theorem describes the full off-diagonal range.
- In infinite dimension, every pair of cardinalities $0,1,2,\dots,\infty$ occurs, so even the maximally incompatible pairs $(\infty,0)$ and $(0,\infty)$ are realized by normal operators.
- A normal operator cannot pair an invertible corner with a compact corner; the obstruction is an essential-numerical-range argument, and it forces any such extreme asymmetry to use a non-invertible, non-compact corner such as a quasiaffinity.
- Finite extreme examples are structurally rigid: they must have $2m$ distinct eigenvalues and be cyclic, and they stay at rank-distance at least $\lfloor(m-1)/2\rfloor$ from every operator with property (CR).
Reading between the lines
- A natural testable extension is to replace 'normal' by 'unitary' or another operator class and ask which rank pairs occur; the paper's block-projection method likely transfers, and the answer may differ because unitary corners obey different trace identities.
- The complete finite-dimensional answer raises the inverse problem of describing, for a fixed normal operator $D$, the set of corner-rank pairs produced as the projection varies; the distance bound in Theorem 2.8 suggests this set encodes spectral geometry.
- The open compact-normal question in Section 4 could be approached by testing weighted-shift-type normal operators with infinite-dimensional eigenspaces; if such an operator exists, it would settle almost-reductivity in the negative, and if not, the obstruction would be a genuinely new compactness phenomenon.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the ranks of the off-diagonal corners of a normal operator D with respect to an orthogonal projection P. Writing D = [D1 D2; D3 D4] relative to H = ran P ⊕ ran(I-P), the paper asks which pairs (rank D2, rank D3) are achievable. In finite dimension n, the main theorem (Theorem 2.5) asserts that every pair (j,k) with 1 ≤ j,k ≤ floor(n/2) is achievable, and that this is optimal. The construction reduces to the case rank D2 = m and rank D3 = 1 in dimension 2m (Theorem 2.3), proved via a Hadamard-product rank computation and a Toeplitz matrix. In infinite dimension, Theorem 3.2 asserts that all pairs (j,k) with 0 ≤ j,k ≤ ∞ are achievable. Additional results show that one corner can be a quasiaffinity while the other has prescribed finite rank (Theorems 3.5 and Corollary 3.6), that the pair (D2 invertible, D3 compact) is impossible (Proposition 3.3), and that in certain extremal cases the normal operator must be cyclic (Theorem 2.8). The paper also proposes the notion of almost-reductive operators and poses questions about compact normal operators.
Significance. The finite-dimensional result completely closes the rank-pair realization problem for normal matrices, and the infinite-dimensional constructions are explicit and checkable. The paper is self-contained apart from standard base cases from the authors' earlier work [4] and Radjavi's theorem [6]; no circularity or internal inconsistency affects the main claims. The introduction of almost-reductive operators and the open questions in Section 4 are likely to stimulate further work. The main proofs are constructive and the rank computations are verifiable by hand, which adds to the reliability of the paper.
minor comments (5)
- [Section 3.6, Corollary 3.6] The statement of Corollary 3.6 says that rank (I-P)DP = j and P D(I-P) is a quasiaffinity, but the proof establishes the opposite assignment: D2 = (I-P)DP is a quasiaffinity and D3 = P D(I-P) has rank j. Please correct the statement (or the proof) to match the construction.
- [Section 3.8, proof of Corollary 3.8] The sentence "we may choose a normal operator M ∈ B(H) such that M = [M1 M2; M3 M4], where rank M2 = j and M2 is a quasiaffinity" is contradictory, since a quasiaffinity on an infinite-dimensional space necessarily has infinite rank. Presumably the intended condition is rank M3 = j and M2 a quasiaffinity, consistent with the construction that follows.
- [Section 2.3, proof of Theorem 2.3 (m = 1 case)] The example D = [[1,1],[1,1]] and P = [[1,0],[0,1]] uses the identity projection as P; for this P the off-diagonal corners are 0×0 blocks and have rank 0, not 1. A rank-one projection such as P = [[1,0],[0,0]] would give the desired ranks.
- [Theorem 2.5 and Corollaries 3.6/3.8] The block-matrix displays are ambiguous because zero blocks are omitted; for example, the four-block display in Theorem 2.5 appears to be non-square as printed. Please include explicit zero blocks or describe the block structure as a direct sum so the reader can verify the rank computations.
- [Throughout] There are several minor typographical errors: in the abstract "j, kcan" should read "j, k can"; in the introduction the matrix display is missing a comma; in Corollary 4.7 the summation index is written as j∈N while the summand uses e_n. These should be corrected.
Circularity Check
No significant circularity: the main existence theorems are proved by explicit, self-contained constructions.
full rationale
The paper's central claims are constructive rather than circular. In the finite-dimensional setting, Theorem 2.3 builds A, B, and S explicitly: the diagonal entries α_j = jγ + i, β_k = kγ − i, and S(γ) are chosen so that S • Z = 2iQ has rank one and Sᵗ • Z is a Toeplitz matrix shown to be invertible for γ > 8m, giving exactly rank D₂ = m and rank D₃ = 1. Theorem 2.5 then assembles the general (j,k) case from direct sums of these blocks and verifies the ranks by construction. The only cited base cases are Proposition 3.7, Proposition 3.13, and Theorem 3.15 of the authors' earlier paper [4]; these are prior stated classification theorems about property (CR), not assumptions of the conclusion being proved, and they serve only as low-dimensional starting points. In the infinite-dimensional setting, Theorem 3.2 uses explicit bilateral shifts and tensor products; Theorem 3.5 gives an explicit diagonal M and N and verifies the required identities NAM − MA*N = 0 and injectivity of MAN − NA*M by direct coefficient estimates. Proposition 3.3 invokes Radjavi's theorem [6, Theorem 8] for a negative result, not for the main existence assertions, and it does not smuggle in the desired conclusion. No fitted parameter is relabeled as a prediction, no uniqueness theorem is imported to force a choice, and no ansatz is hidden inside a citation. The open questions in Sections 2.7 and 4.4–4.5 are honestly stated limitations, not circular dependencies.
Assumptions & free parameters
free parameters (1)
- Gamma (gamma), scale parameter in the finite-dimensional construction =
any value > 8m, e.g. 8m+1
assumptions (4)
- standard math Property-(CR) classification and low-dimensional base cases from Livshits, MacDonald, Marcoux and Radjavi [4], including Propositions 3.7 and 3.13 and Theorem 3.15.
- standard math Radjavi's theorem [6, Theorem 8]: for an operator on infinite-dimensional Hilbert space, 0 belongs to the essential numerical range of T*T - TT* plus any compact perturbation.
- standard math Fan-Fong theorem [3, Theorem 1]: a compact hermitian operator H is a self-commutator of a compact operator iff there is an orthonormal basis with zero diagonal entries.
- standard math Wermer's theorem [8]: every compact normal operator on Hilbert space is orthogonally reductive.
invented entities (1)
-
Almost reductive operators (Definition 4.2)
Cite this review
Pith. "Pith review of Normal operators with highly incompatible off-diagonal corners." pith.science (2026). https://pith.science/paper/LXRSHR5A
@misc{pith2026190807024,
author = {Pith},
title = {Pith review of: Normal operators with highly incompatible off-diagonal corners},
year = {2026},
howpublished = {\url{https://pith.science/paper/LXRSHR5A}},
note = {Machine review of arXiv:1908.07024}
}
abstract
Let $\mathcal{H}$ be a complex, separable Hilbert space, and $\mathcal{B}(\mathcal{H})$ denote the set of all bounded linear operators on $\mathcal{H}$. Given an orthogonal projection $P \in \mathcal{B}(\mathcal{H})$ and an operator $D \in \mathcal{B}(\mathcal{H})$, we may write $D=\begin{bmatrix} D_1& D_2 D_3 & D_4 \end{bmatrix}$ relative to the decomposition $\mathcal{H} = \mathrm{ran}\, P \oplus \mathrm{ran}\, (I-P)$. In this paper we study the question: for which non-negative integers $j, k$ can we find a normal operator $D$ and an orthogonal projection $P$ such that $\mathrm{rank}\, D_2 = j$ and $\mathrm{rank}\, D_3 = k$? Complete results are obtained in the case where $\mathrm{dim}\, \mathcal{H} < \infty$, and partial results are obtained in the infinite-dimensional setting.
Reference graph
Works this paper leans on
-
[4]
L. Livshits, G. MacDonald, L.W. Marcoux, and H. Radjavi, Hilbert space operators with compatible off-diagonal corners , J. Funct. Anal. 275 (2018), 892–925
work page 2018
-
[6]
Radjavi, Structure of A∗ A − AA∗ , J
H. Radjavi, Structure of A∗ A − AA∗ , J. Math. Mech. 16 (1966), 19–26
work page 1966
-
[1]
G. Androulakis, A.I. Popov, A. Tcaciuc and V.G. Troitsky , Almost-invariant half-spaces of operators on Banach spaces , Integral Equations Operator Theory 65 (2009), 473–484
work page 2009
-
[2]
R. Bhatia and M.D. Choi, Corners of normal matrices , Proc. Indian Acad. Sci. Math. Sci. 116 (2006), 393–399
work page 2006
-
[3]
P. Fan and C.K. Fong, Which operators are the self-commutators of compact operat ors?, Proc. Amer. Math. Soc. 80 (1980), 58–60
work page 1980
-
[5]
A.I. Popov and A. Tcaciuc, Every operator has almost-invariant subspaces , J. Funct. Anal. 265 (2013), 257–265
work page 2013
-
[7]
Tcaciuc, The invariant subspace problem for rank-one perturbations , Duke Math
A. Tcaciuc, The invariant subspace problem for rank-one perturbations , Duke Math. J. 168 (2019), 1539–1550
work page 2019
-
[8]
Wermer, On invariant subspaces of normal operators , Proc
J. Wermer, On invariant subspaces of normal operators , Proc. Amer. Math. Soc. 3 (1952), 270–277. Department of Pure Mathematics, University of W aterloo, W a terloo, Ontario, CANADA N2L 3G1 E-mail address : LWMarcoux@uwaterloo.ca Department of Pure Mathematics, University of W aterloo, W a terloo, Ontario, CANADA N2L 3G1 E-mail address : HRadjavi@uwaterl...
work page 1952
Reviewed August 14, 2026 · model on record in the stance chip above.
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