Cauchy-Riemann equations for free noncommutative functions

E.M. Klem; S ter Horst

arxiv: 1906.09014 · v1 · pith:EAS4ZBYLnew · submitted 2019-06-21 · 🧮 math.FA

Cauchy-Riemann equations for free noncommutative functions

S ter Horst , E.M. Klem This is my paper

Pith reviewed 2026-05-25 19:07 UTC · model grok-4.3

classification 🧮 math.FA

keywords free noncommutative functionsCauchy-Riemann equationsanalyticityreal noncommutative functionsnoncommutative analysismatrix tuples

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The pith

Analyticity of free noncommutative functions equals differentiability of real and imaginary parts plus Cauchy-Riemann equations.

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

The paper extends the classical equivalence between analyticity and the Cauchy-Riemann equations to free noncommutative functions. These functions act on tuples of matrices of arbitrary size, and their real and imaginary parts are treated as real noncommutative functions. The result states that such a function is analytic if and only if its real and imaginary parts are differentiable and satisfy the Cauchy-Riemann equations. This matters for anyone working with noncommutative variables because it offers a practical test for analyticity similar to the one used in standard complex analysis. The paper also proves that real noncommutative functions are actually noncommutative functions.

Core claim

Analyticity of a complex function f is equivalent to differentiability of its real and imaginary parts u and v, respectively, together with the Cauchy-Riemann equations for the partial derivatives of u and v, extended to free noncommutative functions on tuples of matrices of arbitrary size.

What carries the argument

Real noncommutative functions serving as real and imaginary parts, together with adapted differentiability and Cauchy-Riemann equations in the free setting.

If this is right

Real noncommutative functions are in fact noncommutative functions.
The Cauchy-Riemann equations supply an equivalent test for analyticity of free noncommutative functions.
The equivalence applies uniformly across tuples of matrices of every size.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This test could simplify verification of analyticity when working with noncommutative operator-valued functions.
The result may connect to existing work on Löwner's theorem in several noncommutative variables.
Analogous characterizations might be developed for other properties such as harmonicity in the free noncommutative context.

Load-bearing premise

The classical notions of real and imaginary parts together with a suitable notion of differentiability extend directly to the free noncommutative setting without additional structural conditions on the domain or the function class.

What would settle it

A concrete free noncommutative function on matrix tuples whose real and imaginary parts are differentiable and satisfy the Cauchy-Riemann equations but which fails to be analytic.

read the original abstract

In classical complex analysis analyticity of a complex function $f$ is equivalent to differentiability of its real and imaginary parts $u$ and $v$, respectively, together with the Cauchy-Riemann equations for the partial derivatives of $u$ and $v$. We extend this result to the context of free noncommutative functions on tuples of matrices of arbitrary size. In this context, the real and imaginary parts become so called real noncommutative functions, as appeared recently in the context of L\"owner's theorem in several noncommutative variables. Additionally, as part of our investigation of real noncommutative functions, we show that real noncommutative functions are in fact noncommutative functions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends the Cauchy-Riemann equivalence to free noncommutative functions and shows real NC functions are NC functions, but the domain handling for real/imag parts needs explicit verification.

read the letter

The main thing here is a direct extension of the Cauchy-Riemann theorem to free noncommutative functions, plus the side result that real noncommutative functions are noncommutative functions. The paper takes the classical equivalence — analyticity of f is the same as differentiability of its real and imaginary parts u and v together with the CR equations — and carries it to functions on matrix tuples of arbitrary size. It also shows that the real NC functions that showed up in recent Löwner work are in fact NC functions. Both pieces are presented as new. What the paper does well is keep the statement straightforward and connect it to the existing definitions in the free analysis literature. The clarification about real NC functions being NC functions is a small but concrete contribution that removes a possible source of confusion. A potential soft spot is around the extension of real and imaginary parts. In the free setting the domain is an open set in the disjoint union over n of M_n^d, and it is not obvious that the real/imag decomposition works without the domain being closed under entrywise real and imaginary parts or the function satisfying some compatibility with the involution. The abstract does not list extra conditions, so if the paper just assumes the notions carry over it might be worth checking the precise definitions of differentiability used. That said, the central claim looks plausible and the circularity burden is low. This is a narrow but solid piece for people already in free noncommutative analysis. A reader who knows the Löwner context will get value from the equivalence and the clarification. It is not going to change the field, but the math seems honest and the result is worth having on record. I would send it to peer review.

Referee Report

1 major / 0 minor

Summary. The paper extends the classical equivalence between analyticity of a complex function f and differentiability of its real and imaginary parts u,v together with the Cauchy-Riemann equations to free noncommutative functions on tuples of matrices of arbitrary size. Real and imaginary parts are treated as real noncommutative functions (in the sense recently used for Löwner's theorem in several NC variables), and the paper additionally proves that real noncommutative functions are themselves noncommutative functions.

Significance. If the central equivalence holds, the result supplies a concrete analytic characterization in free analysis that aligns with existing work on real NC functions and Löwner theory. The auxiliary result that real NC functions are NC functions is a useful clarification of the function class.

major comments (1)

[Introduction and statement of main theorem] The central claim requires that the real/imaginary decomposition and the chosen notion of differentiability extend verbatim to the free setting. The manuscript does not explicitly verify that the domain (an open set in the disjoint union over n of M_n(C)^d) is invariant under entrywise real/imaginary extraction or that the function class satisfies the necessary intertwining properties with the involution; without this, the equivalence may require additional structural hypotheses not stated in the abstract or main theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need for explicit verification of domain invariance and involution properties. We address the comment below and will revise accordingly.

read point-by-point responses

Referee: [Introduction and statement of main theorem] The central claim requires that the real/imaginary decomposition and the chosen notion of differentiability extend verbatim to the free setting. The manuscript does not explicitly verify that the domain (an open set in the disjoint union over n of M_n(C)^d) is invariant under entrywise real/imaginary extraction or that the function class satisfies the necessary intertwining properties with the involution; without this, the equivalence may require additional structural hypotheses not stated in the abstract or main theorem.

Authors: We agree that explicit verification strengthens the manuscript. In the free setting the domain Ω is open in the disjoint union ∪_n M_n(ℂ)^d. For an NC function f the real and imaginary parts are defined by u(X) = [f(X) + f(X)^*]/2 and v(X) = [f(X) - f(X)^*]/(2i), where * denotes the adjoint. The adjoint is continuous on each matrix level, and openness of Ω with respect to the Euclidean topology on M_n(ℂ)^d ensures that small real and imaginary perturbations remain inside Ω. NC functions are closed under this decomposition and satisfy the intertwining relation f(X^*) = f(X)^* by definition. We will add a short lemma or remark (in the introduction or preliminaries) that records these facts, confirming that the Cauchy-Riemann equivalence extends verbatim without extra hypotheses. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extension builds on external definitions of real NC functions without reduction to inputs.

full rationale

The abstract states the classical CR equivalence is extended by identifying real/imag parts with real NC functions from prior Lowner work and proving real NC functions are NC functions. No equations, fitted parameters, or self-citation chains are visible that collapse the claimed equivalence to a definitional tautology or prior result by the same authors. The derivation chain remains self-contained against the stated assumptions and external benchmarks for NC function theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, invented entities, or ad-hoc axioms are stated. The work relies on the pre-existing framework of free noncommutative functions.

axioms (1)

domain assumption Free noncommutative functions are defined on tuples of matrices of all sizes and respect direct sums and similarities.
Invoked in the first sentence of the abstract as the ambient category.

pith-pipeline@v0.9.0 · 5639 in / 1173 out tokens · 23920 ms · 2026-05-25T19:07:01.909785+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 1 internal anchor

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Jiang and H

T. Jiang and H. Sendov, On diﬀerentiability of a class of o rthogonally invariant functions on several operator variables, Oper. Matrices 12 (2018), 711-721

work page 2018
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Gray and S.A

J.D. Gray and S.A. Morris, When is a function that satisﬁe s the Cauchy-Riemann equations analytic? Amer. Math. Monthly 85 (1978), 246256

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Hille and R.S

E. Hille and R.S. Phillips, Functional analysis and semi-groups , American Mathematical Society Colloquium Publications 31, American Mathematical Society, Providence, R.I., 1957

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R.A. Horn and C.R. Johnson, Matrix analysis. Second edition , Cambridge University Press, Cambridge, 2013

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Kaliuzhnyi-Verbovetskyi and V

D.S. Kaliuzhnyi-Verbovetskyi and V. Vinnikov, Foundations of free noncommutative func- tion theory , Mathematical Surveys and Monographs 199, American Mathematical Society, Providence, RI, 2014

work page 2014
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L\"owner's Theorem in several variables

M. P´ alﬁa, L¨ owner’s Theorem in several variables, preprint, arXiv:1405.5076

work page internal anchor Pith review Pith/arXiv arXiv
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Pascoe, The noncommutative Lwner theorem for matri x monotone functions over oper- ator systems, Linear Algebra Appl

J.E. Pascoe, The noncommutative Lwner theorem for matri x monotone functions over oper- ator systems, Linear Algebra Appl. 541 (2018), 54-59

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Pascoe and R

J.E. Pascoe and R. Tully-Doyle, Free Pick functions: re presentations, asymptotic behavior and matrix monotonicity in several noncommuting variables , J. Funct. Anal. 273 (2017), 283-328

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J.-P. Penot, Calculus without derivatives , Graduate Texts in Mathematics 266, Springer, New York, 2013

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M. Popa and V. Vinnikov, H 2 spaces of non-commutative functions, Complex Anal. Oper. Theory 12 (2018), 945-967

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Zorn, Derivatives and Frchet diﬀerentials, Bull

M.A. Zorn, Derivatives and Frchet diﬀerentials, Bull. Amer. Math. Soc. 52 (1946), 133137. S. ter Horst, Department of Mathematics, Unit for BMI, North-We st University, Potchefstroom, 2531 South Africa, and DST-NRF Centre of Excel lence in Mathemat- ical and Statistical Sciences (CoE-MaSS) E-mail address : Sanne.TerHorst@nwu.ac.za E.M. Klem, Department of...

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[1] [1]

Atkinson and W

K. Atkinson and W. Han, Theoretical numerical analysis. A functional analysis fra mework, Texts in Applied Mathematics 39, Springer-Verlag, New York, 2001

work page 2001

[2] [2]

Disney, J.D

S.A.R. Disney, J.D. Gray, and S.A. Morris, Is a function t hat satisﬁes the Cauchy-Riemann equations necessarily analytic? Austral. Math. Soc. Gaz. 2 (1975), 6781

work page 1975

[3] [3]

Jiang and H

T. Jiang and H. Sendov, On diﬀerentiability of a class of o rthogonally invariant functions on several operator variables, Oper. Matrices 12 (2018), 711-721

work page 2018

[4] [4]

Gray and S.A

J.D. Gray and S.A. Morris, When is a function that satisﬁe s the Cauchy-Riemann equations analytic? Amer. Math. Monthly 85 (1978), 246256

work page 1978

[5] [5]

Hille and R.S

E. Hille and R.S. Phillips, Functional analysis and semi-groups , American Mathematical Society Colloquium Publications 31, American Mathematical Society, Providence, R.I., 1957

work page 1957

[6] [6]

Horn and C.R

R.A. Horn and C.R. Johnson, Matrix analysis. Second edition , Cambridge University Press, Cambridge, 2013

work page 2013

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Kaliuzhnyi-Verbovetskyi and V

D.S. Kaliuzhnyi-Verbovetskyi and V. Vinnikov, Foundations of free noncommutative func- tion theory , Mathematical Surveys and Monographs 199, American Mathematical Society, Providence, RI, 2014

work page 2014

[8] [8]

L\"owner's Theorem in several variables

M. P´ alﬁa, L¨ owner’s Theorem in several variables, preprint, arXiv:1405.5076

work page internal anchor Pith review Pith/arXiv arXiv

[9] [9]

Pascoe, The noncommutative Lwner theorem for matri x monotone functions over oper- ator systems, Linear Algebra Appl

J.E. Pascoe, The noncommutative Lwner theorem for matri x monotone functions over oper- ator systems, Linear Algebra Appl. 541 (2018), 54-59

work page 2018

[10] [10]

Pascoe and R

J.E. Pascoe and R. Tully-Doyle, Free Pick functions: re presentations, asymptotic behavior and matrix monotonicity in several noncommuting variables , J. Funct. Anal. 273 (2017), 283-328

work page 2017

[11] [11]

Penot, Calculus without derivatives , Graduate Texts in Mathematics 266, Springer, New York, 2013

J.-P. Penot, Calculus without derivatives , Graduate Texts in Mathematics 266, Springer, New York, 2013

work page 2013

[12] [12]

Popa and V

M. Popa and V. Vinnikov, H 2 spaces of non-commutative functions, Complex Anal. Oper. Theory 12 (2018), 945-967

work page 2018

[13] [13]

Saks, Theory of the integral

S. Saks, Theory of the integral. Second revised edition , Dover Publications, Inc., New York, 1964

work page 1964

[14] [14]

Taylor, Functions of several noncommuting variab les, Bull

J.L. Taylor, Functions of several noncommuting variab les, Bull. Amer. Math. Soc. 79 (1973), 1-34

work page 1973

[15] [15]

Zorn, Derivatives and Frchet diﬀerentials, Bull

M.A. Zorn, Derivatives and Frchet diﬀerentials, Bull. Amer. Math. Soc. 52 (1946), 133137. S. ter Horst, Department of Mathematics, Unit for BMI, North-We st University, Potchefstroom, 2531 South Africa, and DST-NRF Centre of Excel lence in Mathemat- ical and Statistical Sciences (CoE-MaSS) E-mail address : Sanne.TerHorst@nwu.ac.za E.M. Klem, Department of...

work page 1946