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bibkey: johnson2026powersroots
authors: C. R. Johnson; S. Mondal; M. J. Tsatsomeros
year: 2026
title: "Powers, Roots, and Positivity"
doi: 10.48550/arXiv.2609.06298
url: https://arxiv.org/abs/2609.06298v1
claim: "The entries marked Open indicate cases for which preservation under Pick functions remains an open question."
strata_touched:
  - D5/S3/Quantum/Matrix/InfinitelyDivisiblePickRefutation
  - D5/S3/Quantum/Matrix/EmbeddablePickRefutation
license: citation-only
triage: anchor

Locator

Author text: https://arxiv.org/abs/2609.06298v1.

• DOI: https://doi.org/10.48550/arXiv.2609.06298

C. R. Johnson, S. Mondal, and M. J. Tsatsomeros, Powers, Roots, and Positivity, arXiv:2609.06298v1, Definition 2.7, Lemma 2.8, Remark 2.9, Definition 3.25, and Table 2, p. 22.

Pick functions and strong infinite divisibility

Definition 2.7 states verbatim:

Let D := C(−∞, 0]. A complex function f analytic on D is called a Pick function if ℑf(z) ℑz ≥ 0, ∀z ∈ D

Definition 3.25 states verbatim:

A nonnegative matrix A ∈ Mn(R) is said to be an infinitely divisible matrix (IDM) if, for every positive integer m, there exists a nonnegative matrix Km such that (Km)^m = A. If, in addition, A is invertible, then A is called a strongly infinitely divisible matrix (SIDM).

Table 2, “Behavior of Matrix Classes under Pick Functions and Fractional Powers”, p. 22, row 12 reads verbatim:

Strongly Infinitely Divisible Matrix (SIDM) | Open | Yes

Remark 3.28, p. 17, states verbatim:

The embedding problem, introduced in [11], asks whether a given stochastic matrix A can be realized as the transition matrix of a continuous-time Markov process. It was shown in [24] that a nonsingular stochastic matrix A is embeddable if and only if A = e^B for some essentially nonnegative matrix B satisfying Be = 0.

Table 2, “Behavior of Matrix Classes under Pick Functions and Fractional Powers”, p. 22, row 13 reads verbatim:

Embeddable Matrix | Open | Yes

The note below Table 2 states verbatim:

The entries marked Open indicate cases for which preservation under Pick functions remains an open question.

The framing question in §1 is verbatim:

Given A ∈ C and a Pick function f, is f[A] ∈ C?

Lemma 2.8 represents a Pick function by

$$f(z)=a+bz+{\int }_0^{\infty }\frac{tz-1}{t+z}d\mu (t),a\in \mathbb{R},b\ge 0,\mu \ge 0.$$

Remark 2.9 gives, for spectrum in $D$,

$$f[A]=aI+bA+{\int }_0^{\infty }\left(tI-(1+t^2)(A+tI{)}^{-1}\right)d\mu (t).$$

Repository-derived result

The companion theorem result : ¬ claim refutes preservation restricted to finitely supported positive measures, functions positive on $(0,\infty )$, and upper-triangular SIDM matrices with positive diagonal. Their spectrum is their diagonal, inside $(0,\infty )\subset D$, so Remark 2.9 applies. The general preservation statement implies this restriction; refuting it therefore refutes the general statement. Take $a=1$, $b=0$, and $\mu ={\delta }_1$, so $f(z)=2z/(1+z)$. The pole lies outside $D$, and

$$\mathrm{Im}f(z)=\frac{2\mathrm{Im}z}{|1+z{|}^2}.$$

Let $N$ be the $4\times 4$ shift with ones on its first superdiagonal and put

$$E(s)=I+sN+\frac{s^2}{2}{N}^2+\frac{s^3}{6}{N}^3,A=E(1).$$

The identity $E(s)E(u)=E(s+u)$ supplies nonnegative roots $E(1/m)$ of every positive integer order. The determinant of $A$ is 1, and $A$ is upper triangular with diagonal 1, hence spectrum $\{1\}\subset (0,\infty )\subset D$. Moreover,

$$(A+I{)}^{-1}=\frac{1}{2}I-\frac{1}{4}N+\frac{1}{48}{N}^3,f[A]=I+\frac{1}{2}N-\frac{1}{24}{N}^3.$$

Thus $f[A{]}_{1,4}=-1/24$, and $f[A]$ is not nonnegative and hence not SIDM. The answer to Table 2, row 12, is No. The Lean module proves the real-axis positivity, roots, determinant 1, upper triangularity, positive diagonal, inverse identity, and negative entry; complex analyticity and the displayed imaginary-part identity are mathematical explanations of the source representation, not separately formalized there.

Neighboring references

The source’s Table 2, rows 7 and 13, concerns semipositive and embeddable matrices. Neither row is settled by the companion theorem.

ASSUMED-UNVERIFIED

The supplied quotations and locators bind to arXiv:2609.06298v1; their agreement with the author text is ASSUMED-UNVERIFIED. No exhaustive literature or priority claim is established. The finite-measure encoding uses Lemma 2.8 and Remark 2.9; their general analytic statements are cited rather than formalized.