On Indecomposable triples associated with nilpotent operators

Ahmed El Khantach; El Hassan Zerouali

arxiv: 1906.09523 · v1 · pith:A5DFJVQZnew · submitted 2019-06-23 · 🧮 math.FA

On Indecomposable triples associated with nilpotent operators

Ahmed El Khantach , El Hassan Zerouali This is my paper

Pith reviewed 2026-05-25 18:13 UTC · model grok-4.3

classification 🧮 math.FA

keywords indecomposable triplesnilpotent operatorsinvariant subspacesclassificationnilpotency indexlinear algebra constructions

0 comments

The pith

Indecomposable triples (V, T, U) for nilpotent T and invariant U are completely classified when n_U=1 for arbitrary p and when n_U=2 with n_V at most 3.

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

The paper studies triples consisting of a finite-dimensional space V, a nilpotent operator T on V, and a T-invariant subspace U. It sets n_U equal to the dimension of the kernel of T restricted to U. The goal is to classify those triples that cannot be written as a direct sum of two smaller nontrivial triples of the same kind. Classifications are obtained that depend on the nilpotency index p of T together with the values of n_U and the dimension of V. Explicit lists are supplied in the cases n_U=1 (any p) and n_U=2 with dimension of V at most 3, while the situation for p at least 6 is shown to contain infinitely many indecomposables.

Core claim

Complete classifications of indecomposable triples are given for arbitrary p when n_U=1, and when n_U=2 and n_V ≤ 3. The case p ≥ 6, where the number of indecomposable triples is infinite, is also investigated. The case p ≤ 5 is recaptured by constructive proofs based on linear algebra tools.

What carries the argument

The indecomposable triple (V, T, U) with U T-invariant, [U] defined as ker(T restricted to U), and the triple admitting no nontrivial direct-sum decomposition into smaller triples of the same form.

If this is right

For every nilpotency index p the indecomposable triples with n_U=1 form a finite explicit list.
When n_U=2 and dim V ≤ 3 the indecomposable triples also form an explicit finite list.
For p ≥ 6 there exist infinitely many distinct indecomposable triples.
The classifications for small p are recovered directly from linear-algebra constructions without external results.

Where Pith is reading between the lines

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

The explicit lists for small n_U may be used to enumerate all possible lattices of invariant subspaces arising from such triples.
The transition to infinitely many indecomposables at p=6 indicates that any attempt at a uniform closed-form description must change character once the nilpotency index exceeds five.

Load-bearing premise

The definition of indecomposability for the triple (V,T,U) is taken as the standard one in the literature on nilpotent operators and invariant subspaces.

What would settle it

An explicit indecomposable triple with n_U=1 whose form lies outside the listed families for a given p would show that the classification is incomplete.

read the original abstract

We consider in this paper the family of triples $(V, T, U),$ where $ V$ is a finite dimensional space, $T $ is a nilpotent linear operator on $V$ and $U $ is an invariant subspace of $T$. Denote $[U]= ker(T_{|U})$, and $n_U= dim([U] )$. Our main goal is to investigate possible classification of indecomposable triples. The obtained classification depends on the order of nilpotency $p$, on $n_U$ and on $n_V$. Complete classifications are given for arbitrary $p$, when $n_U=1$, and when $n_U=2$ and $n_V \le 3$. The case $ p \le 5$, treated in \cite{ring} is recaptured by using constructive proofs based on linear algebra tools. The case $p\ge 6$, where the number of indecomposable triples is infinite, is also investigated.

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 classifications of indecomposable nilpotent triples to arbitrary p at n_U=1 and n_U=2 with n_V≤3 using constructive linear-algebra proofs, but the lists' exhaustiveness still needs explicit verification that no extra splittings exist under the standard definition.

read the letter

The paper classifies indecomposable triples (V, T, U) where T is nilpotent and U is invariant. It recaptures the p ≤ 5 cases from the cited reference with direct constructions based on bases and dimension counts, and it adds the cases of arbitrary p when n_U = 1 plus n_U = 2 with n_V ≤ 3. For p ≥ 6 it notes the infinite families and describes them. The constructive approach is the clearest gain: readers get explicit matrix forms or invariant descriptions instead of abstract existence arguments, which makes the objects usable for building examples or testing decompositions. The linear-algebra tools keep the arguments self-contained and avoid any fitted parameters or circular steps. The main soft spot is completeness. The central claim requires that every indecomposable triple falls into one of the listed families and that the invariants (p, n_U, n_V) catch every possible direct-sum splitting allowed by the definition. The abstract invokes the standard definition from the literature, so the concern is not fatal, but the paper must show that no additional indecomposables arise outside those families; without that verification the lists could be incomplete. The citation pattern is clean: the prior work is treated as external input. This paper is for specialists in finite-dimensional nilpotent operator theory who need concrete lists for small n_U and n_V. A reader working on invariant-subspace problems or example construction will find usable families. It is narrow but the extensions are new and the method is checkable, so it deserves a serious referee who can verify the constructions and the exhaustiveness arguments.

Referee Report

1 major / 1 minor

Summary. The manuscript classifies indecomposable triples (V, T, U) consisting of a finite-dimensional vector space V, a nilpotent operator T on V, and a T-invariant subspace U. It supplies complete classifications for arbitrary nilpotency index p when n_U = dim(ker(T|_U)) = 1, and when n_U = 2 with n_V ≤ 3, via explicit linear-algebra constructions. The p ≤ 5 cases from prior work are recovered constructively, and the p ≥ 6 regime (where infinitely many indecomposables exist) is examined.

Significance. The constructive linear-algebra approach yields explicit families together with dimension-count arguments, which is a strength for direct verification and reproducibility. If the listed families are exhaustive, the work extends the literature on nilpotent operators with invariant subspaces by furnishing concrete descriptions in the low-n_U cases and clarifying the transition to infinitely many indecomposables.

major comments (1)

[Abstract and the sections containing the classification statements] The central claim of completeness for the cases n_U=1 (arbitrary p) and n_U=2 with n_V≤3 rests on the assertion that every indecomposable triple is captured by one of the constructed families. The manuscript invokes the standard definition of indecomposability but must supply an explicit argument (e.g., via the invariants p, n_U, n_V and a proof that no undetected direct-sum splittings exist) showing that the constructions are exhaustive; without this, the classification remains incomplete.

minor comments (1)

The notation [U] := ker(T|_U) is introduced early; verify that it is used uniformly in all subsequent statements and proofs.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the suggestion to make the completeness argument more explicit. We address the major comment below.

read point-by-point responses

Referee: [Abstract and the sections containing the classification statements] The central claim of completeness for the cases n_U=1 (arbitrary p) and n_U=2 with n_V≤3 rests on the assertion that every indecomposable triple is captured by one of the constructed families. The manuscript invokes the standard definition of indecomposability but must supply an explicit argument (e.g., via the invariants p, n_U, n_V and a proof that no undetected direct-sum splittings exist) showing that the constructions are exhaustive; without this, the classification remains incomplete.

Authors: We agree that an explicit argument establishing exhaustiveness is required for a complete classification. In the revised version we will add a dedicated subsection that uses the invariants p, n_U and n_V to prove that every indecomposable triple must be isomorphic to one of the constructed families. The argument proceeds by assuming a nontrivial direct-sum decomposition of the triple and deriving a contradiction with indecomposability (via the definition and the given dimension constraints) unless the triple matches one of the listed types. This will be placed immediately after the construction of the families for the n_U=1 and n_U=2, n_V≤3 cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity; classifications derived from explicit linear-algebra constructions.

full rationale

The paper derives its classifications of indecomposable triples (V, T, U) through direct linear-algebra constructions, dimension counts, and explicit families for the cases n_U=1 (arbitrary p) and n_U=2 with n_V≤3. It recaptures the p≤5 results from the external citation via new constructive proofs rather than treating the citation as a load-bearing premise. No self-definitional reductions, fitted inputs renamed as predictions, or uniqueness claims imported from self-citations appear in the derivation chain. The work is self-contained against external benchmarks via standard definitions and constructive methods.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the standard axioms of finite-dimensional linear algebra over an algebraically closed field (implicitly C or algebraically closed) and the usual definition of nilpotency and invariance; no free parameters or new entities are introduced.

axioms (2)

standard math Finite-dimensional vector spaces over an algebraically closed field admit Jordan canonical form for nilpotent operators.
Invoked implicitly when classifying nilpotent actions via chains or partitions of dimensions.
domain assumption Indecomposability of the triple is defined by the non-existence of a direct-sum decomposition respecting both T and U.
The classification target is defined in terms of this standard notion from the literature on operator triples.

pith-pipeline@v0.9.0 · 5698 in / 1501 out tokens · 25363 ms · 2026-05-25T18:13:12.258345+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

M., and Schmidmeier, M

Ringel, C. M., and Schmidmeier, M. (2008). Invariant sub spaces of nilpotent linear operators, I. Journal fur die reine und angewandte Mathematik (Crelles Journal), 2008(614), 1-52

work page 2008
[2]

Rodman and H

R Bru, L. Rodman and H. Schneider, Extensions of Jordan ba ses for invariant subspaces of a matrix. Linear Algebra and its Applications, 150 (1991) 209 -225

work page 1991
[3]

Faouzi, On the orbit of invariant subspaces of linear o perators in ﬁnite-dimensional spaces, Linear Algebra and its Applications 329 (2001) 171-174

A. Faouzi, On the orbit of invariant subspaces of linear o perators in ﬁnite-dimensional spaces, Linear Algebra and its Applications 329 (2001) 171-174

work page 2001
[4]

Richman and E

F. Richman and E. A. W alker, Subgroups of p5 -bounded grou ps, in: Abelian groups and modules, Trends Math., Birkhauser, Basel, (1999), 5573. A. Elkhantach, Center of Mathematical research in Rabat. Moh amed V University in Rabat. F aculty of Sciences. BP 1014 Rabat Morocco E. H. Zerouali, Center of Mathematical research in Rabat. Moh amed V University i...

work page 1999

[1] [1]

M., and Schmidmeier, M

Ringel, C. M., and Schmidmeier, M. (2008). Invariant sub spaces of nilpotent linear operators, I. Journal fur die reine und angewandte Mathematik (Crelles Journal), 2008(614), 1-52

work page 2008

[2] [2]

Rodman and H

R Bru, L. Rodman and H. Schneider, Extensions of Jordan ba ses for invariant subspaces of a matrix. Linear Algebra and its Applications, 150 (1991) 209 -225

work page 1991

[3] [3]

Faouzi, On the orbit of invariant subspaces of linear o perators in ﬁnite-dimensional spaces, Linear Algebra and its Applications 329 (2001) 171-174

A. Faouzi, On the orbit of invariant subspaces of linear o perators in ﬁnite-dimensional spaces, Linear Algebra and its Applications 329 (2001) 171-174

work page 2001

[4] [4]

Richman and E

F. Richman and E. A. W alker, Subgroups of p5 -bounded grou ps, in: Abelian groups and modules, Trends Math., Birkhauser, Basel, (1999), 5573. A. Elkhantach, Center of Mathematical research in Rabat. Moh amed V University in Rabat. F aculty of Sciences. BP 1014 Rabat Morocco E. H. Zerouali, Center of Mathematical research in Rabat. Moh amed V University i...

work page 1999