arxiv: 2605.01397 · v1 · submitted 2026-05-02 · 🧮 math.FA

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Weak Minimizing Property and the Compact Perturbation Property for the Minimum Modulus

Anselmo Raposo Jr., Geivison Ribeiro

Pith reviewed 2026-05-10 15:18 UTC · model grok-4.3

classification 🧮 math.FA

keywords minimum moduluscompact perturbationc0non-attaining operatorsBanach space operatorsweak minimizing property

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The pith

The pair (c₀, c₀) fails the compact perturbation property for the minimum modulus.

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

The paper shows that (c₀, c₀) does not satisfy the compact perturbation property for the minimum modulus, or CPPm. This property would mean that if an operator T fails to attain its minimum modulus m(T), then no compact perturbation can make that value larger. The authors construct an explicit operator T on c₀ that does not attain m(T), yet a rank-one compact operator K exists with m(T + K) > m(T). They further prove that the same failure occurs for any space of the form 𝕂 ⊕_∞ Y when Y is non-reflexive and the pair is taken with itself.

Core claim

We give a negative answer to this question by proving that (c₀, c₀) does not have the CPPm. The proof is constructive, exhibiting a non-min-attaining operator whose minimum modulus is strictly increased by a rank-one compact perturbation. Moreover, if X=𝕂⊕_∞Y with Y non-reflexive, then the pair (X,X) fails the CPPm.

What carries the argument

The explicit construction of a non-min-attaining operator T on c₀ (or on X=𝕂⊕_∞Y) such that there exists a rank-one compact K with m(T+K) > m(T).

If this is right

(c₀, c₀) fails the CPPm.
If X = 𝕂 ⊕_∞ Y with Y non-reflexive, then (X, X) fails the CPPm.
The failure is tied to the non-reflexivity of the underlying space and the explicit sequence-space construction.

Where Pith is reading between the lines

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

Analogous failures of the CPPm may hold in other non-reflexive Banach spaces that lack the direct-sum structure used here.
The same constructive approach could be tested on related perturbation properties for operators between different sequence spaces.

Load-bearing premise

There exists a specific non-min-attaining operator on c₀ whose minimum modulus increases under some rank-one compact perturbation, due to the sequence-space properties of c₀.

What would settle it

Direct verification on the constructed operator that m(T+K) equals m(T) for the given rank-one compact K, or that no compact perturbation increases m(T) for any non-attaining T on c₀.

read the original abstract

For an operator $T:X\to Y$, denote $m(T)=\inf\{\|Tx\|:x\in S_X\}$. A sequence $(x_n)$ in $S_X$ is said to be minimizing for $T$ if $\|Tx_n\|\to m(T)$. The weak minimizing property (WmP), introduced by Chakraborty, requires that every operator admitting a non-weakly null minimizing sequence attains its minimum modulus. More recently, Han~\cite{Han2026} introduced the Compact Perturbation Property for the minimum modulus (CPPm), which requires that for every operator $T:X\to Y$ that does not attain its minimum modulus, \[ \sup_{K\in\mathcal{K}(X,Y)} m(T+K)=m(T). \] In~\cite{Han2026}, it is shown that $(\ell_1,\ell_1)$ fails both properties, while $(c_0,c_0)$ fails the WmP. However, whether $(c_0,c_0)$ has the CPPm was left open (Problem~3.6). In this paper, we give a negative answer to this question by proving that $(c_0,c_0)$ does not have the CPPm. The proof is constructive, exhibiting a non-min-attaining operator whose minimum modulus is strictly increased by a rank-one compact perturbation. Moreover, we show that this phenomenon is not specific to $c_0$: if $X=\mathbb{K}\oplus_\infty Y$ with $Y$ non-reflexive, then the pair $(X,X)$ fails the CPPm.

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 constructive counterexample showing (c0,c0) fails the CPPm and answers Han's open problem negatively, with a generalization to certain non-reflexive spaces.

read the letter

The paper settles the question of whether (c0, c0) has the Compact Perturbation Property for the minimum modulus. It does not, and the authors show this with a specific operator T from c0 to c0 that fails to attain its min modulus m(T), but there is a rank one compact operator K such that m(T + K) is larger than m(T). They also prove that this happens more generally whenever the space is an l-infinity sum of the field with a non-reflexive Banach space. This is new because it supplies the first explicit negative answer to the open problem left in Han's 2026 paper. The construction is not in the prior literature, and it uses a direct approach rather than non-constructive methods. The paper does well by keeping the focus on the counterexample and explaining how the sequence space nature of c0 allows the perturbation to increase the minimum modulus. The generalization part shows that the failure is tied to non-reflexivity in a natural way. Soft spots are small. The main assumption is that the chosen T and K satisfy the strict inequality in the minimum modulus, which depends on careful choice of sequences in c0 that are minimizing but allow the perturbation to help. As long as the full proof checks the limits and norms correctly, there is no issue. The stress test found no internal inconsistency, and I agree that the outline looks standard for this area. A reader working on perturbation properties or minimum attaining operators in Banach spaces would get value from this. It is targeted at that subfield and helps map out where these properties hold or fail. The paper shows clear thinking in addressing the open question head on with a construction. It deserves a serious referee. I would recommend sending it for peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the pair (c₀, c₀) fails the Compact Perturbation Property for the minimum modulus (CPPm). It does so via an explicit construction of a non-minimum-attaining operator T : c₀ → c₀ for which there exists a rank-one compact K with m(T+K) > m(T). The authors further show that if X = 𝕂 ⊕_∞ Y with Y non-reflexive, then the pair (X, X) likewise fails CPPm. This supplies a negative answer to the open Problem 3.6 of Han (2026).

Significance. The result is significant because it resolves an open question with a concrete, constructive counterexample rather than a non-constructive existence argument. The use of standard sequence-space techniques on c₀ and the exploitation of non-reflexivity in the direct-sum generalization provide clear, falsifiable insight into when CPPm fails. This strengthens the literature on minimum-modulus properties by distinguishing the behavior of CPPm from the already-known failure of the weak minimizing property on the same pair.

minor comments (2)

[Main construction section] In the statement of the main theorem (presumably Theorem 3.1 or its equivalent), the explicit sequences or basis vectors used to define T and the rank-one perturbation K should be written with full index ranges to facilitate direct verification of the inequality m(T+K) > m(T).
[Generalization paragraph] The generalization to X = 𝕂 ⊕_∞ Y would benefit from a short remark clarifying whether the same rank-one form of K works verbatim or requires a minor adjustment when Y is an arbitrary non-reflexive space.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its significance in providing a constructive counterexample to the open Problem 3.6, and the recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; constructive counterexample is self-contained

full rationale

The paper resolves the open Problem 3.6 from Han2026 by exhibiting an explicit non-min-attaining operator T on c0 (and its generalization to X = K ⊕_∞ Y) together with a rank-one compact K such that m(T+K) > m(T). This is a direct, parameter-free construction relying on sequence-space properties of c0 and non-reflexivity of Y; no equation reduces to a prior fitted quantity, no ansatz is smuggled via self-citation, and the citation to Han2026 serves only to state the open question rather than to justify the proof. The derivation chain is therefore independent of its inputs and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper introduces no free parameters, new entities, or ad-hoc axioms; it relies on standard, previously established properties of the spaces c₀ and non-reflexive Banach spaces.

axioms (2)

domain assumption c₀ is a non-reflexive Banach space with the standard Schauder basis and known dual properties.
Invoked to construct the operator T and the rank-one perturbation that increases m(T).
domain assumption Existence of non-min-attaining operators on c₀ with non-weakly-null minimizing sequences.
Used as the starting point for the counterexample; drawn from prior literature on the minimum modulus.

pith-pipeline@v0.9.0 · 5602 in / 1481 out tokens · 61572 ms · 2026-05-10T15:18:03.900347+00:00 · methodology

discussion (0)

Reference graph

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