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The Brown Measure of Non-Hermitian Sums of Projections

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Pith's one-line read For $p,q$ freely independent Hermitian operators with two-point spectra, the paper computes the Brown measure of $X=p+iq$ in closed form: four corner atoms plus a continuous part carried by an explicit density on a hyperbola-arc.

desk verdict Main theorem is likely correct and the hyperbola/atom results are genuinely useful, but the key Stieltjes transform in Proposition 5.6 has a fixable algebraic error and the normalization of the density is asserted rather than checked. read the letter →

arxiv 2411.13804 v2 pith:FUNGS227 submitted 2024-11-21 math.OA math.PR

Max Sun Zhou This is my paper · ORCID
classification math.OAmath.PR MSC 46L5460B2046L10
keywords Brownmeasurefreeprobabilityprojectionsnon-normaloperatorshyperbolasupportvonNeumannalgebrasS-transformempiricalspectraldistribution
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper computes, in closed form, the Brown measure of $X = p + iq$ when $p$ and $q$ are freely independent Hermitian operators whose spectra each consist of two points. The measure is a convex combination of four point masses sitting at the four corner sums of the atoms, plus a continuous part whose density is written explicitly. The continuous part lives on the arc of a hyperbola inside a rectangle whose geometry depends only on the atom positions, while its density depends only on the two atom weights. Such operators are never normal, so this is a genuinely non-Hermitian family, and the formula gives a concrete, fully explicit instance of the Brown-measure machinery beyond the earlier R-diagonal examples.

What carries the argument

The engine is the model of the von Neumann algebra generated by two projections: its nontrivial part is isomorphic to $M_2(L^\infty(\nu))$ with trace $\mathbb{E}_\nu[\tfrac12\operatorname{tr}]$, so a general element becomes a matrix-valued function on $(0,\pi/2)$, and the two projections become explicit $2\times2$ matrix functions. In this model the element $x=pqp+(1-p)(1-q)(1-p)$ becomes a scalar diagonal matrix, so $\nu$ is identified as its spectral measure. Freeness enters through the $\psi$- and $S$-transforms: the paper computes $\psi_{pqp}$ and $\psi_{(1-p)(1-q)(1-p)}$, removes the atomic contributions of $p\wedge q$ and $(1-p)\wedge(1-q)$, and obtains the Stieltjes transform of $\nu$; the change of variables $t=\cos^2\theta$ converts it into the stated density. The same model turns $e(z-X)e$ into a matrix-valued function, and its singular values feed the formula for $\log\Delta(z-X)$; taking the distributional Laplacian splits the Brown measure into the four atoms and the pushforward curves.

What would settle it

Take a concrete case such as $a=b=1/2$, $\alpha=0$, $\alpha'=1$, $\beta=0$, $\beta'=4/5$, form independently rotated finite matrices $P_n,Q_n$ with these spectra, and compare the empirical spectral distribution of $P_n+iQ_n$ with the density given by Theorem 6.1. If the limiting empirical distribution does not concentrate on the predicted hyperbola-arc segment with the predicted density, the formula is wrong.

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Extended reading notes

Core claim

The central theorem asserts that if $\mu_p=a\delta_\alpha+(1-a)\delta_{\alpha'}$ and $\mu_q=b\delta_\beta+(1-b)\delta_{\beta'}$, with $a,b\in(0,1)$ and $\alpha\neq\alpha'$, $\beta\neq\beta'$, then the Brown measure of $X=p+iq$ is $$\mu=\epsilon_{00}\delta_{\$\alpha$+i\$\beta$}+\epsilon_{01}\delta_{\$\alpha$+i\$\beta$'}+\epsilon_{10}\delta_{\$\alpha$'+i\$\beta$}+\epsilon_{11}\delta_{\$\alpha$'+i\$\beta$'}+\epsilon\,\mu',$$ where $\epsilon_{00}=\max(0,a+b-1)$, $\epsilon_{01}=\max(0,a-b)$, $\epsilon_{10}=\max(0,b-a)$, $\epsilon_{11}=\max(0,1-a-b)$, and $\epsilon=1-(\epsilon_{00}+\epsilon_{01}+\epsilon_{10}+\epsilon_{11})>0$. The continuous measure $\mu'$ is the average of the pushforwards of $\nu$ under two explicit curves $\lambda_1,\lambda_2$ that parameterize the intersection of the hyperbola $$H=\left\{(x,y):\left(x-\frac{\$\alpha$+\$\alpha$'}2\right)^2-\left(y-\frac{\$\beta$+\$\beta$'}2\right)^2=\frac{(\$\alpha$'-\$\alpha$)^2-(\$\beta$'-\$\beta$)^2}4\right\}$$ with the rectangle $$R=\{x+iy: x\in[\$\alpha$\wedge\$\alpha$',\$\alpha$\vee\$\alpha$'],\ y\in[\$\beta$\wedge\$\beta$',\$\beta$\vee\$\beta$']\}.$$ The measure $\nu$ has density $$\frac{d\nu}{d\$\theta$}=\frac{2}{\pi\epsilon}\operatorname{Im}\sqrt{f(\$sec^{2}$\$\theta$)}\,\cot\$\theta$,\qquad f(x)=1+(4ab-2(a+b))x+(a-b)^$2x^{2}$,$$ with the square root of a negative number taken on the positive imaginary axis. The paper also proves that the continuous part is always present and that the map from the laws of $p,q$ to the Brown measure is injective.

Load-bearing premise

The computation rests on assuming that the corner of the algebra left over after intersecting the two projections is faithfully described by the $2\times2$ matrix model, and that the parameter measure in that model is exactly the spectral measure of $x=pqp+(1-p)(1-q)(1-p)$ as computed through freeness.

Editorial extensions

If this is right

  • The Brown measure is always supported on the hyperbola-arc $H\cap R$, and it fills the whole arc exactly when $a=b=1/2$.
  • The measure has 0, 1, or 2 atoms, with masses $|a+b-1|$ and $|a-b|$; the continuous part always has positive total mass, so the measure is never purely atomic.
  • Every operator $X=p+iq$ of this form is non-normal, so the closed-form result is a genuinely non-Hermitian Brown-measure computation.
  • The Brown measure determines the laws of $p$ and $q$: the assignment $(\mu_p,\mu_q)\mapsto\mu_X$ is injective.
  • The continuous part is invariant under swapping $p$ and $q$, and replacing $p$ or $q$ by its complement reverses the parameter direction of the density.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: before the announced convergence proof appears, the explicit density can serve as a direct numerical benchmark: simulate finite matrices with the stated spectra and compare the empirical spectral distribution of $P_n+iQ_n$ with the formula.
  • Editorial extension: because any compactly supported law can be approximated by two-atom laws, weak limits of these hyperbola-arc supports give a concrete prediction for the Brown measure of a sum of general freely independent self-adjoint operators, a case the paper does not treat.
  • Editorial extension: the same two-projection model should yield explicit Brown measures for affine combinations such as $ap+bq$ or $p+cq$ with complex $c$, since only the matrix function $e(p+iq)e$ changes in the computation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper computes the Brown measure of X = p + iq, where p and q are freely independent Hermitian operators with two-point spectra. The computation uses the classical model of the von Neumann algebra generated by two projections as M2(L∞(ν)) after cutting by the central projection e, together with free-probability transforms to determine the weights of the four atoms and the density of the continuous part ν. The main result (Theorem 6.1) expresses the Brown measure as a convex combination of four atoms and a continuous measure μ′ supported on a hyperbola intersected with a rectangle, with explicit formulas for the density and the curve parameterization. The paper also derives qualitative properties: atoms appear only at the four corners, the continuous part is absolutely continuous, the measure has various symmetries, and the map from the laws of p and q to the Brown measure is injective.

Significance. If correct, this is a valuable explicit computation in a non-R-diagonal, non-normal setting, adding to a sparse list of tractable Brown-measure examples. The paper is self-contained and uses a coherent strategy (spectral measure of H_z, Fuglede–Kadison determinant, distributional Laplacian) that is standard and well executed in most places. The resulting measure is explicit enough to read off atoms, density, support, and symmetries, and the paper provides numerical simulations as illustrations. The two-projection model is a standard tool and its application here is appropriate. However, the proof of the key density formula in Proposition 5.6 contains an algebraic error in the Stieltjes transform, and the normalization of the proposed density is asserted rather than verified. These issues are local and fixable, but they are load-bearing for Theorem 6.1.

major comments (2)
  1. [Section 5, Proposition 5.6, Eq. (129)] The displayed formula for Gν∗(z) is algebraically incorrect. Applying the identity G(z) = (1/z)(ψ(1/z)+1) to the expression for ψ_exe in (127) gives Gν∗(z) = sqrt(f(1/z))/(τ(e)(z−1)) + (1 − 1/τ(e))/z − C/(τ(e)z(z−1)), not the expression shown, which is missing the +1/z term. Consequently, the displayed Gν∗ decays like O(1/z²) and violates the standard normalization (24) of a Stieltjes transform of a probability measure. The missing term is analytic on (0,1), so the subsequent density formula (134) is not directly invalidated, but the derivation as written is not valid. The algebra must be corrected.
  2. [Section 5, Proposition 5.6, after Eq. (134)] The paper's only justification that the density in (121) integrates to 1 is the parenthetical remark that ν is 'a probability measure, being the spectral measure of a non-zero element of eMe'. In light of the algebraic error in (129), this assertion needs independent verification. The corrected Stieltjes transform should be shown to satisfy (24), or the integral of the density in (121) should be computed directly. This is essential because the total mass of ν enters the coefficient ϵ in Theorem 6.1, and a wrong normalization would change the Brown measure.
minor comments (5)
  1. [Lemma 4.4, Eqs. (74)–(75)] The equivalence 'z ∈ R ⇔ (73) ≤ 0 ⇔ x ∈ [α∧α′, α∨α′] or y ∈ [β∧β′, β∨β′]' uses 'or' where the intended condition is 'and'. On the hyperbola the two conditions are in fact equivalent to each other, but the statement as written is false in general and the proof of (77) employs an invalid implication. The presentation should be corrected to use the conjunction, which is what the rest of the argument relies on.
  2. [Sections 3 and 4] There are unresolved cross-references: Section 3 says 'In Section??, we discuss some further work' and the proofs of Propositions 4.1 and 4.2 refer to 'Section??' for the two-projection model. These placeholders should be filled in.
  3. [Proposition 4.2 proof] The phrase 'Fortherestoftheproof, assumethat' is missing spaces (a LaTeX typo).
  4. [Corollary 6.5(2)] The statement 'f is either quadratic with positive leading coefficient or f is linear with negative slope' should be justified when a = 1 − b and f is quadratic; the argument that f has a single root at 1 follows from Proposition 5.1, but this is not explicitly connected in the proof.
  5. [Theorem 6.1, notation of ϵij] The indices of the ϵij may confuse readers because they correspond to the spectral projections p′ = χ_{α′}(p) and q′ = χ_{β′}(q); a short parenthetical reminder of this correspondence (as in the proof, after Proposition 5.3) would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Brown measure is computed from the input laws of p and q through the two-projection model and free-probability transforms, with no fitted quantity or self-citation chain used as the load-bearing step.

full rationale

The derivation is self-contained rather than circular. Section 4 derives the general form of the Brown measure as four atoms plus the pushforward of the central measure ν under the eigenvalue branches λ1, λ2 (Proposition 4.3), starting from the definition of Brown measure, the standard external two-projection model (7)-(9), and a direct computation of the spectral measure of Hz(X). The parameters are not guessed or fitted to the Brown measure: the weights τ(eij) and τ(e) are computed in Proposition 5.3 from the free independence of the projections via ψpqp and Lemma 5.2, and ν is computed in Proposition 5.6 by inverting the Stieltjes transform of exe, obtained by decomposing ψpqp and ψ(1−p)(1−q)(1−p). No equation is substituted back into the Brown measure to fix a free constant, and no parameter is calibrated to a subset of the target data and then renamed a prediction. The only load-bearing citations are to standard external results, especially Voiculescu's two-projection model [14] and Mingo-Speicher [10]; these are not self-citations of the author. Even if the algebraic concern raised about equation (129) were valid, that would be a correctness issue rather than circularity, because the displayed transform is derived from the input data and not chosen to reproduce the final measure. The derivation therefore does not reduce to its own inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central derivation uses standard free probability and operator algebra tools, not fitted parameters. The main structural input is the two-projection model, which is cited from prior work.

assumptions (6)
  • domain assumption Brown measure is given by 1/(2π) ∇² log Δ(z-X).
    Definition 2.2; this is the standard definition used throughout.
  • domain assumption The von Neumann algebra generated by two projections has the model M2(L∞((0,1),ν*)) with explicit matrix representations for p and q.
    Section 2.2, equations (7)-(8); result from Voiculescu [14].
  • standard math S-transform multiplicativity for free multiplicative convolution of pqp.
    Theorem 2.12; used in Proposition 5.1.
  • domain assumption Freeness of p and q implies freeness of their spectral projections p′ and q′.
    Section 5, paragraph after equation (93); standard free probability fact.
  • standard math Moment method: compact measures on R are determined by their moments.
    Used in Propositions 4.1 and 4.2.
  • domain assumption The traces τ(eij) are given by the max(0, ·) formulas for free projections.
    Proposition 5.3; derived using non-tangential limits.

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Cite this review

Pith. "Pith review of The Brown Measure of Non-Hermitian Sums of Projections." pith.science (2026). https://pith.science/paper/FUNGS227

@misc{pith2026241113804,
  author       = {Pith},
  title        = {Pith review of: The Brown Measure of Non-Hermitian Sums of Projections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FUNGS227}},
  note         = {Machine review of arXiv:2411.13804}
}
abstract

We compute the Brown measure of the non-normal operators $X = p + i q$, where $p$ and $q$ are Hermitian, freely independent, and have spectra consisting of $2$ atoms. The computation relies on the model of the non-trivial part of the von Neumann algebra generated by 2 projections as $2 \times 2$ random matrices. We observe that these measures are supported on hyperbolas and note some other properties related to their atoms and symmetries.

Figures

Figures reproduced from arXiv: 2411.13804 by the authors.

Figure 1
Figure 1. ESD of Xn = Pn + iQn µPn = (1/2)δ0 + (1/2)δ1 µQn = (1/2)δ0 + (1/2)δ4/5 n = 1000 Motivated by the hyperbola and rectangle appearing in Corollary 4.5, we introduce the definition of the hyperbola and rectangle associated with X: Definition 4.6. Let p, q ∈ (M, τ ) be Hermitian with laws: (86) µp = aδα + (1 − a)δα′ µq = bδβ + (1 − b)δβ′ . Let A = α ′ − α and B = β ′ − β. The hyperbola associated with X is (87) H = ( z =… view at source ↗
Figure 2
Figure 2. illustrates the behavior in Corollary 6.4 where the Brown measure of X = p + iq is approximated by the ESD of Xn = Pn + iQn for deterministic Pn, Qn ∈ Mn(C). Note the symmetry between the two ESDs. (a) ESD of Xn = Pn + iQn µPn = (4/5)δ0 + (1/5)δ9/10 µQn = (1/5)δ0 + (4/5)δ1 n = 1000 (b) ESD of Xn = Pn + iQn µPn = (1/5)δ0 + (4/5)δ9/10 µQn = (1/5)δ0 + (4/5)δ1 n = 1000 [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗
Figure 3
Figure 3. ESDs for Corollary 6.5 Finally, we conclude that the Brown measure of X = p + iq uniquely determines the laws of p and q. Here we allow p and q to be possibly constant. Corollary 6.6. Let p, q ∈ (M, τ ) be Hermitian and freely independent and (150) µp = aδα + (1 − a)δα′ µq = bδβ + (1 − b)δβ′ . where a, b ∈ [0, 1] and α, α′ , β, β′ ∈ R. Let µ be the Brown measure of X = p + iq. Then, the assignment (µp, µq) 7→ µ is 1… view at source ↗

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Convergence of the Laws of Non-Hermitian Sums of Projections

    math.OA 2024-11 conditional novelty 7.0 of 10

    For random matrices X_n = P_n + iQ_n with P_n and Q_n independently Haar-rotated Hermitian two-atom matrices, the empirical spectral distribution converges almost surely to the Brown measure of p + iq whenever the law...

  2. Quaternionic Green's Function and the Brown Measure of Atomic Operators

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    For free atomic Hermitian operators p and q, the boundary heuristic for the Brown measure of X = p + iq implies the boundary is an algebraic curve, and the paper provides an explicit resultant-based algorithm to produ...

Reference graph

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