Linear preservers of rank k projections

Lucijan Plevnik

arxiv: 2512.10645 · v2 · submitted 2025-12-11 · 🧮 math.FA

Linear preservers of rank k projections

Lucijan Plevnik This is my paper

Pith reviewed 2026-05-16 22:58 UTC · model grok-4.3

classification 🧮 math.FA

keywords linear preserversrank k projectionsWigner's theoremself-adjoint operatorsfinite rank operatorsHermitian matricesunitary matricesHilbert space

0 comments

The pith

Linear maps on self-adjoint finite-rank operators that preserve rank-k projections are fully characterized, extending Wigner's theorem.

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

The paper characterizes linear maps on the real vector space of self-adjoint finite-rank bounded operators on a complex Hilbert space that preserve the set of rank-k projections. This generalizes Wigner's theorem from rank-one projections to higher ranks. The approach first solves a related problem for linear maps on trace-zero 2k by 2k Hermitian matrices that preserve unitary matrices. It also examines maps sending rank-k projections to finite-rank projections, proves they map to fixed rank, gives complete description when the Hilbert space dimension is two, and provides examples showing the general case is more complex.

Core claim

We characterize linear maps on F_s(H) which preserve the set of all rank k projections by reducing the problem to maps on the space of trace zero (2k) x (2k) Hermitian matrices that preserve the subset of unitary matrices. For maps from F_s(H) to F_s(K) sending projections of rank k to finite rank projections, they send rank k projections to projections of a fixed rank, with complete description when dim H = 2.

What carries the argument

Linear maps preserving the set of rank k projections on the real vector space F_s(H) of self-adjoint finite-rank operators; the key reduction is to unitary preservers in the space H_{0,2k} of trace-zero Hermitian matrices.

If this is right

Such maps send rank k projections to projections of a fixed rank.
The complete form of the maps is given when the dimension of the Hilbert space is two.
The characterization reduces to first describing unitary-preserving linear maps on trace-zero 2k by 2k Hermitian matrices.
In higher dimensions the maps exhibit more complicated behavior as shown by examples.

Where Pith is reading between the lines

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

If the characterization holds, it suggests that symmetry preservers for higher-rank projections behave similarly to unitary or anti-unitary conjugations as in the rank-one case.
This connects preserver problems for projections to matrix theory over low-dimensional Hermitian spaces.
Testing the maps in dimensions greater than two could reveal whether additional forms exist beyond the characterized ones.

Load-bearing premise

The maps are assumed to be linear over the reals on the vector space of self-adjoint finite-rank bounded operators.

What would settle it

A counterexample would be a linear map on F_s(H) that preserves the set of rank-k projections but cannot be expressed in the forms given by the characterization for the given dimension.

read the original abstract

Let $\mathcal H$ be a complex Hilbert space and $\mathcal F_s (\mathcal H)$ the real vector space of all self-adjoint finite rank bounded operators on $\mathcal H$. We generalize the famous Wigner's theorem by characterizing linear maps on $\mathcal F_s (\mathcal H)$ which preserve the set of all rank $k$ projections. In order to do this, we first characterize linear maps on the real vector space $\mathcal H_{0, 2k}$ of trace zero $(2k) \times (2k)$ hermitian matrices which preserve the subset of unitary matrices in $\mathcal H_{0, 2k}$. We also study linear maps from $\mathcal F_s (\mathcal H)$ to $\mathcal F_s (\mathcal K)$ sending projections of rank $k$ to finite rank projections. We prove some properties of such maps, e.g. that they send rank $k$ projections to projections of a fixed rank. We give the complete description of such maps in the case $\dim \mathcal H = 2$. We give several examples which show that in the general case the problem to describe all such maps seems to be complicated.

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 a clean auxiliary result on trace-zero Hermitian matrices and a full classification only for dim H=2, while the general case stays partial with counterexamples.

read the letter

The main new piece is the characterization of real-linear maps on the 2k by 2k trace-zero Hermitian matrices that preserve the self-adjoint unitaries, plus the explicit form of the rank-k projection preservers when dim H equals 2. Those parts look like genuine additions to the Wigner-preserver literature and are worth having on record. They also prove some basic properties that hold more generally, such as the image of a rank-k projection under these maps being a projection of fixed rank, and they supply counterexamples that show why the general problem is messy. That evidence is concrete and useful for anyone who might try to push further. The soft spot is the gap between the stated goal and what is actually proved. The abstract presents a characterization for arbitrary H, yet the text only finishes the job in dimension 2 and gives only partial information plus counterexamples for larger or infinite-dimensional spaces. The matrix-level result is not shown to lift cleanly without extra assumptions, so the central claim for general H rests on an incomplete argument. This paper is for specialists in linear preserver problems and operator theory who already know the classical Wigner results. A reader looking for the dim-2 case or the matrix auxiliary theorem will get something concrete; someone hoping for a full general theorem will be disappointed. It deserves a serious referee because the new matrix result and the low-dimensional classification are solid enough to check and because the counterexamples clarify the boundary of what is known.

Referee Report

2 major / 1 minor

Summary. The paper claims to generalize Wigner's theorem by characterizing real-linear maps on the space F_s(H) of self-adjoint finite-rank operators on a complex Hilbert space H that preserve the set of all rank-k projections. It reduces the problem to classifying linear maps on the trace-zero Hermitian 2k x 2k matrices that preserve the self-adjoint unitaries therein, provides a complete explicit description only when dim H = 2, and studies maps from F_s(H) to F_s(K) sending rank-k projections to finite-rank projections, proving they map to projections of fixed rank while supplying counter-examples and partial properties indicating that the general case is complicated.

Significance. If the full characterization for arbitrary H were established, the work would meaningfully extend classical linear-preserver results (Wigner, Kadison, etc.) to finite-rank settings and could inform questions in quantum information and operator theory. The matrix-level classification and the dim-H=2 case constitute concrete progress, but the absence of a lifting argument from the finite-dimensional unitary preserver to general (possibly infinite-dimensional) H restricts the result's scope and immediate applicability.

major comments (2)

[Abstract] Abstract and opening paragraphs: the stated goal is a characterization of linear maps on F_s(H) for general H, yet the manuscript supplies an explicit form only for dim H=2 and explicitly notes that the general case 'seems to be complicated,' providing only partial properties (constant-rank output) and counter-examples without a derivation showing how the 2k x 2k unitary-preserver result lifts to arbitrary H.
[Introduction / reduction paragraph] The reduction step (characterizing maps on trace-zero Hermitian 2k x 2k matrices preserving self-adjoint unitaries) is presented as the key tool for the general-H case, but no argument is given establishing that every linear preserver on F_s(H) restricts to or is determined by such a matrix map when dim H > 2 or H is infinite-dimensional.

minor comments (1)

[Section on matrix case] Notation: the space H_{0,2k} is introduced without an explicit statement of its inner-product or norm, which would clarify the linearity assumption.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We agree that the abstract and introduction should more precisely delineate the scope of the results, and we will revise accordingly to avoid overstating the generality. Below we respond point by point to the major comments.

read point-by-point responses

Referee: [Abstract] Abstract and opening paragraphs: the stated goal is a characterization of linear maps on F_s(H) for general H, yet the manuscript supplies an explicit form only for dim H=2 and explicitly notes that the general case 'seems to be complicated,' providing only partial properties (constant-rank output) and counter-examples without a derivation showing how the 2k x 2k unitary-preserver result lifts to arbitrary H.

Authors: We accept this criticism. The explicit characterization via the matrix result is complete only for dim H=2; for general (including infinite-dimensional) H we establish auxiliary properties such as constant output rank and supply counter-examples indicating the problem is more involved. We will revise the abstract and opening paragraphs to state clearly that we (i) characterize the trace-zero Hermitian 2k x 2k unitary preservers, (ii) give the full description when dim H=2, and (iii) obtain partial results together with counter-examples for the general case, without claiming a complete lifting. revision: yes
Referee: [Introduction / reduction paragraph] The reduction step (characterizing maps on trace-zero Hermitian 2k x 2k matrices preserving self-adjoint unitaries) is presented as the key tool for the general-H case, but no argument is given establishing that every linear preserver on F_s(H) restricts to or is determined by such a matrix map when dim H > 2 or H is infinite-dimensional.

Authors: The reduction is used directly only when the underlying space is 2k-dimensional (hence for the complete dim-H=2 case). For larger or infinite-dimensional H the action on orthogonal complements and the possible non-local nature of the maps prevent an automatic restriction, which is precisely why we describe the general problem as complicated and limit ourselves to the constant-rank property and counter-examples. We will insert a clarifying paragraph after the reduction statement explaining this limitation and that a full lifting remains open. revision: partial

standing simulated objections not resolved

Absence of a general lifting argument from the 2k-dimensional matrix classification to arbitrary (including infinite-dimensional) Hilbert spaces; providing such an argument would require new techniques outside the present work.

Circularity Check

0 steps flagged

No circularity; standard reduction to finite-dimensional preserver problem is independent of target result.

full rationale

The derivation proceeds by first characterizing linear maps on the 2k-dimensional trace-zero Hermitian matrices that preserve the self-adjoint unitaries, then lifting to maps on F_s(H) that preserve rank-k projections. This is a conventional linear-preserver reduction that does not define the output set in terms of itself, does not rename a fitted quantity as a prediction, and invokes no self-citation chain for the uniqueness or form of the maps. The explicit classification is supplied only for dim H=2 while general-H results remain partial, but neither step reduces the claimed characterization to its own inputs by construction. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard axioms of Hilbert-space operator theory and linear algebra; no free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Standard properties of complex Hilbert spaces and bounded self-adjoint operators
Invoked throughout the setup of F_s(H) and rank-k projections.

pith-pipeline@v0.9.0 · 5496 in / 1114 out tokens · 23237 ms · 2026-05-16T22:58:12.351996+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We first characterize linear maps on the real vector space H_{0,2k} of trace zero (2k)×(2k) hermitian matrices which preserve the subset of unitary matrices in H_{0,2k}.
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.13 ... ϕ(A) = s U A U^*, A ∈ H_{0,2k}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

B¨ ottcher and I

MR 1477662 [BS10] A. B¨ ottcher and I. M. Spitkovsky,A gentle guide to the basics of two projections theory, Linear Algebra Appl.432(2010), no. 6, 1412–1459. MR 2580440 [Fau02] Claude-Alain Faure,An elementary proof of the fundamental theorem of projective ge- ometry, Geom. Dedicata90(2002), 145–151. MR 1898158 [Gal08] A. Gal´ antai,Subspaces, angles and ...

work page 2010
[2]

[Gv16] Gy¨ orgy P´ al Geh´ er and PeterˇSemrl,Isometries of Grassmann spaces, J. Funct. Anal. 270(2016), no. 4, 1585–1601. MR 3447720 [Jor75] Camille Jordan,Essai sur la g´ eom´ etrie ` andimensions, Bull. Soc. Math. France3(1875), 103–174. MR 1503705 [Mol01] Lajos Moln´ ar,Transformations on the set of alln-dimensional subspaces of a Hilbert space preser...

work page 2016
[3]

MR 2267033 [Mol08] Lajos Moln´ ar,Maps on then-dimensional subspaces of a Hilbert space preserving prin- cipal angles, Proc. Amer. Math. Soc.136(2008), no. 9, 3205–3209. MR 2407085 [Pan19] Mark Pankov,Wigner’s type theorem in terms of linear operators which send projections of a fixed rank to projections of other fixed rank, J. Math. Anal. Appl.474(2019),...

work page 2008

[1] [1]

B¨ ottcher and I

MR 1477662 [BS10] A. B¨ ottcher and I. M. Spitkovsky,A gentle guide to the basics of two projections theory, Linear Algebra Appl.432(2010), no. 6, 1412–1459. MR 2580440 [Fau02] Claude-Alain Faure,An elementary proof of the fundamental theorem of projective ge- ometry, Geom. Dedicata90(2002), 145–151. MR 1898158 [Gal08] A. Gal´ antai,Subspaces, angles and ...

work page 2010

[2] [2]

[Gv16] Gy¨ orgy P´ al Geh´ er and PeterˇSemrl,Isometries of Grassmann spaces, J. Funct. Anal. 270(2016), no. 4, 1585–1601. MR 3447720 [Jor75] Camille Jordan,Essai sur la g´ eom´ etrie ` andimensions, Bull. Soc. Math. France3(1875), 103–174. MR 1503705 [Mol01] Lajos Moln´ ar,Transformations on the set of alln-dimensional subspaces of a Hilbert space preser...

work page 2016

[3] [3]

MR 2267033 [Mol08] Lajos Moln´ ar,Maps on then-dimensional subspaces of a Hilbert space preserving prin- cipal angles, Proc. Amer. Math. Soc.136(2008), no. 9, 3205–3209. MR 2407085 [Pan19] Mark Pankov,Wigner’s type theorem in terms of linear operators which send projections of a fixed rank to projections of other fixed rank, J. Math. Anal. Appl.474(2019),...

work page 2008