Invertible positive maps that are not automorphism

K. C. Sivakumar; Pavankumar Raickwade

arxiv: 2605.14739 · v1 · pith:A265BKO7new · submitted 2026-05-14 · 🧮 math.FA

Invertible positive maps that are not automorphism

Pavankumar Raickwade , K. C. Sivakumar This is my paper

Pith reviewed 2026-06-30 19:54 UTC · model grok-4.3

classification 🧮 math.FA

keywords positive linear mapscone automorphismsrank one perturbationinvertible operatorsordered vector spacesfunctional analysis

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The pith

Every cone automorphism admits a rank-one perturbation that remains positive and invertible but whose inverse does not preserve the cone.

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

The paper gives an explicit construction, valid whenever a cone in a real normed space meets one of three mild conditions, that produces an invertible linear map sending the cone into itself while its inverse sends some cone vector outside the cone. The construction begins with any automorphism of the cone and adds a rank-one correction chosen so the new map stays positive and bijective. This separates the property of being positive and invertible from the stronger property of being a cone automorphism. The same method supplies concrete examples in four different ordered spaces.

Core claim

In a real normed vector space X equipped with a cone K that is either closed with nonempty interior, possesses nonzero extreme rays, or is closed when X is Banach, every cone automorphism S admits a rank-one perturbation T = S + uv* that is invertible, satisfies T(K) ⊆ K, yet fails to satisfy T^{-1}(K) ⊆ K.

What carries the argument

A rank-one perturbation of a given cone automorphism, chosen so the resulting operator stays positive and bijective while its inverse ceases to be positive.

If this is right

Positive invertibility is strictly weaker than being a cone automorphism under the stated conditions on K.
The same perturbation technique works uniformly across spaces with interior points, extremal rays, or Banach-space completeness.
Concrete counterexamples exist in four distinct ordered vector spaces.
The construction supplies an explicit family of maps that are positive and invertible without being automorphisms.

Where Pith is reading between the lines

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

Similar rank-one adjustments might separate positivity from other algebraic properties such as complete positivity in operator algebras.
The result suggests that numerical checks for cone automorphisms must verify the inverse separately rather than relying on invertibility alone.
The method could be tested on cones without interior or extreme rays to see where the construction fails.

Load-bearing premise

The cone must be closed with nonempty interior, or have nonzero extreme rays, or be closed inside a Banach space.

What would settle it

An explicit computation in a concrete space (for instance the positive orthant in R^2) showing that every rank-one perturbation of the identity either loses positivity, loses invertibility, or keeps a positive inverse.

read the original abstract

Let $X$ be a real normed vector space with a cone $K\subseteq X$ satisfying either (i) $K$ is closed with non-empty interior or (ii) $K$ has non-zero extremals or (iii) $K$ is closed and $X$ is a Banach space. In this short note, we provide a method to construct an invertible linear map $T\colon X\to X$ such that $T[K]\subseteq K$ but $T^{-1}[K]\not\subseteq~K$. In particular, we show that, for every cone automorphism $S\colon X\to X$, there exists a rank one perturbation of $S$ which is positive and invertible, but does not have a positive inverse. We provide examples from four diverse situations.

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 gives an explicit rank-one perturbation turning any cone automorphism into a positive invertible map whose inverse is not positive, under three standard cone conditions.

read the letter

The main contribution is a direct construction: start with a cone automorphism S, add a rank-one term chosen using the cone's properties, and get T positive and invertible while T^{-1} fails positivity. They handle three cases for the cone K—closed with interior, nonzero extremals, or closed in a Banach space—and check the claim with four concrete examples.

The construction is spelled out clearly enough that the stress-test note confirms they pick the vectors and functionals from the cone conditions and verify the inverse formula directly. That part holds up without hidden steps.

The soft spots are limited. It is a short note, so it stays at the level of existence via perturbation rather than classifying all such maps or giving applications. Novelty is plausible from the abstract but would need the references checked to confirm it does not overlap prior perturbation results. The three conditions cover many standard settings but leave out some exotic cones.

This is for researchers in ordered vector spaces or positive operators who want explicit counterexamples to the idea that positive invertibility implies positive inverse. A reader looking for concrete constructions in functional analysis would find it useful.

It deserves peer review because the argument is explicit and the examples make the claim falsifiable.

Referee Report

0 major / 2 minor

Summary. The paper claims that if X is a real normed space and K a cone satisfying one of three conditions (closed with nonempty interior; possessing nonzero extremals; or closed with X Banach), then for any cone automorphism S there exists a rank-one perturbation T of S such that T is positive and invertible but T^{-1} is not positive. An explicit construction is supplied for each case, together with four concrete examples.

Significance. The explicit rank-one perturbation formulas and the four worked examples constitute the main contribution; they furnish concrete, verifiable instances separating the notions of positive invertibility and cone automorphism under minimal hypotheses on K. The result is local to the listed cone conditions and does not claim necessity of those conditions.

minor comments (2)

The title phrase “not automorphism” is grammatically incomplete; “not an automorphism” or “not automorphisms” would be clearer.
In the abstract the notation T[K] and T^{-1}[K] is used without prior definition; a single sentence recalling that these denote the image of the cone would aid readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; explicit construction independent of inputs

full rationale

The paper supplies an explicit rank-one perturbation construction for any given cone automorphism S, using the stated cone conditions (closed with interior, non-zero extremals, or Banach) solely to select vectors/dual functionals that preserve positivity and invertibility while violating positivity of the inverse via the rank-one update formula. No fitted parameters, no self-definitional equations, no load-bearing self-citations, and no renaming of known results. The central claim is a direct existence proof verified by examples in four settings; the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the three listed cone conditions stated in the abstract; no free parameters or invented entities are introduced in the given text.

axioms (1)

domain assumption X is a real normed vector space and K is a cone satisfying one of the three listed conditions (closed with nonempty interior, nonzero extremals, or closed in a Banach space).
Explicitly required in the abstract as the setting in which the construction holds.

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discussion (0)

Reference graph

Works this paper leans on

10 extracted references

[1]

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and Tourky, Rabee.Cones and duality, volume 84 ofGrad

Aliprantis, Charalambos D. and Tourky, Rabee.Cones and duality, volume 84 ofGrad. Stud. Math.Provi- dence, RI: (AMS), 2007

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[3]

G. P. Barker. Theory of cones.Linear Alg. Appl., 39:263-291, 1981

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Berman and R

A. Berman and R. J. Plemmons.Nonnegative Matrices in the Mathematical Sciences, volume 9 ofClassics in Applied Mathematics, SIAM, 1994

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[5]

Gl¨ uck and M

J. Gl¨ uck and M. R. Weber. Almost interior points in ordered Banach spaces and the long-term behaviour of strongly positive operator semigroups.Stud. Math., 254(3):237-263, 2013

2013
[6]

M. S. Gowda, R. Sznajder, and J. Tao. The automorphism group of a completely positive cone and its lie algebra.Linear Alg. Appl., 438(10):3862–3871, 2013

2013
[7]

J. Horne. On the automorphism group of a cone.Linear Algebra Appl., 21(2):111–121, 1978

1978
[8]

Kalauch and O

A. Kalauch and O. van Gaans.Pre-Riesz spaces,volume 66 ofDe Gruyter Expo. Math.Berlin: De Gruyter, 2019

2019
[9]

Y. Shitov. Linear mappings preserving the copositive cone.Proc. Amer. Math. Soc., 149(8):3173–3176, 2021

2021
[10]

Sznajder

R. Sznajder. A representation theorem for the lorentz cone automorphisms.J. Optim. Theory Appl., 202(1):296–302, July 2024. Department of Mathematics, Indian Institute of Technology Madras, Chennai, Tamil Nadu, India Email address:prraickwade@gmail.com Department of Mathematics, Indian Institute of Technology Madras, Chennai, Tamil Nadu, India Email addre...

2024

[1] [1]

C. D. Aliprantis and O. Burkinshaw.Positive operators, volume 119 ofPure Appl. Math., Academic Press, New York, 1985

1985

[2] [2]

and Tourky, Rabee.Cones and duality, volume 84 ofGrad

Aliprantis, Charalambos D. and Tourky, Rabee.Cones and duality, volume 84 ofGrad. Stud. Math.Provi- dence, RI: (AMS), 2007

2007

[3] [3]

G. P. Barker. Theory of cones.Linear Alg. Appl., 39:263-291, 1981

1981

[4] [4]

Berman and R

A. Berman and R. J. Plemmons.Nonnegative Matrices in the Mathematical Sciences, volume 9 ofClassics in Applied Mathematics, SIAM, 1994

1994

[5] [5]

Gl¨ uck and M

J. Gl¨ uck and M. R. Weber. Almost interior points in ordered Banach spaces and the long-term behaviour of strongly positive operator semigroups.Stud. Math., 254(3):237-263, 2013

2013

[6] [6]

M. S. Gowda, R. Sznajder, and J. Tao. The automorphism group of a completely positive cone and its lie algebra.Linear Alg. Appl., 438(10):3862–3871, 2013

2013

[7] [7]

J. Horne. On the automorphism group of a cone.Linear Algebra Appl., 21(2):111–121, 1978

1978

[8] [8]

Kalauch and O

A. Kalauch and O. van Gaans.Pre-Riesz spaces,volume 66 ofDe Gruyter Expo. Math.Berlin: De Gruyter, 2019

2019

[9] [9]

Y. Shitov. Linear mappings preserving the copositive cone.Proc. Amer. Math. Soc., 149(8):3173–3176, 2021

2021

[10] [10]

Sznajder

R. Sznajder. A representation theorem for the lorentz cone automorphisms.J. Optim. Theory Appl., 202(1):296–302, July 2024. Department of Mathematics, Indian Institute of Technology Madras, Chennai, Tamil Nadu, India Email address:prraickwade@gmail.com Department of Mathematics, Indian Institute of Technology Madras, Chennai, Tamil Nadu, India Email addre...

2024