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arxiv: 2604.06625 · v1 · submitted 2026-04-08 · 🧮 math.FA

The Bishop-Phelps-Bollob\'as property for the numerical radius: a Zizler-type approach

Sun Kwang Kim , Han Ju Lee , Miguel Martin , Oscar Roldan This is my paper

Pith reviewed 2026-05-10 18:22 UTC · model grok-4.3

classification 🧮 math.FA
keywords Bishop-Phelps-Bollobás propertynumerical radiusBanach spacesℓ_∞direct sumscompact operatorsZizler reformulation
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The pith

Real ℓ_∞ satisfies the Bishop-Phelps-Bollobás property for the numerical radius while the complex space ℓ_1 ⊕_∞ c_0 does not.

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

The authors apply a Zizler-type reformulation to the Bishop-Phelps-Bollobás property for the numerical radius. They prove that the real Banach space ℓ_∞ satisfies this property. In contrast, the complex Banach space ℓ_1 ⊕_∞ c_0 does not, marking the first natural example without renorming techniques where operators attaining the numerical radius are dense but the property fails. The paper also strengthens previous results on the connection between the property for pairs of spaces and for their direct sums, and studies the Zizler approach for different pairs and for compact operators.

Core claim

Through a Zizler-type perspective, the real Banach space ℓ_∞ satisfies the BPBp-nu. The complex space ℓ_1 ⊕_∞ c_0 does not satisfy the BPBp-nu. This provides the first natural example of a Banach space in which the numerical radius attaining operators are dense but the BPBp-nu fails.

What carries the argument

The Zizler-type reformulation of the BPBp-nu, which recasts the approximation condition in terms of nearby points at which the numerical radius is attained for the operators under study.

If this is right

  • The real Banach space ℓ_∞ satisfies the BPBp-nu.
  • The complex Banach space ℓ_1 ⊕_∞ c_0 fails the BPBp-nu despite the density of numerical radius attaining operators.
  • The 2016 interplay theorems between BPBp for a pair (X,Y) and BPBp-nu for the direct sum X ⊕ Y hold in strengthened form.
  • The Zizler-type BPBp relates to the classical BPBp and to BPBp-nu for a range of Banach space pairs.
  • The same Zizler-type analysis yields results when the operators are restricted to be compact.

Where Pith is reading between the lines

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

  • Direct-sum constructions may systematically produce spaces where density of attaining operators separates from the full BPBp-nu without artificial renorming.
  • The real versus complex distinction appears decisive for whether the Zizler reformulation succeeds on ℓ_∞-type sums.
  • The method could be tested on other standard spaces such as ℓ_p for 1 < p < ∞ to map the boundary between positive and negative cases.
  • Restricting to compact operators opens the possibility of parallel statements for other operator ideals.

Load-bearing premise

The Zizler-type reformulation correctly captures the numerical-radius approximation condition for the real ℓ_∞ and complex ℓ_1 ⊕_∞ c_0 spaces.

What would settle it

Exhibiting a bounded linear operator on the complex ℓ_1 ⊕_∞ c_0 that cannot be approximated in operator norm by any numerical-radius attaining operator would confirm the failure of BPBp-nu there, while a proof that every operator on real ℓ_∞ admits such approximations would confirm the positive result.

read the original abstract

We investigate the Bishop-Phelps-Bollob\'as property for the numerical radius (BPBp-nu) through a Zizler-type perspective on the classical Bishop-Phelps-Bollob\'as property (BPBp). This approach allows us to establish two new results: the real Banach space $\ell_\infty$ satisfies the BPBp-nu, while the complex space $\ell_1 \oplus_\infty c_0$ does not. Note that the latter provides the first natural example (constructed without renorming techniques) of a Banach space where the numerical radius attaining operators are dense but the BPBp-nu fails. Along the way, we strengthen the main results of the paper [Kim et al, On the Bishop-Phelps-Bollob\'as theorem for operators and numerical radius, Studia Math., 2016] concerning the interplay between the BPBp for the pair $(X,Y)$ and the BPBp-nu for a direct sum $X\oplus Y$ of Banach spaces. We further explore the validity of the Zizler-type BPBp across different pairs of Banach spaces, and how this property relates to the classical BPBp and the BPBp-nu. Finally, we specialize our analysis to the framework of compact operators.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a Zizler-type reformulation of the Bishop-Phelps-Bollobás property adapted to the numerical radius (BPBp-nu). Using this, it shows that the real Banach space ℓ_∞ satisfies the BPBp-nu, whereas the complex Banach space ℓ_1 ⊕_∞ c_0 does not. The latter is presented as the first natural example of a space in which numerical-radius-attaining operators are dense but the BPBp-nu fails. The paper strengthens the interplay theorems of Kim et al. (2016) between BPBp for pairs (X, Y) and BPBp-nu for direct sums, explores the Zizler-type property for different pairs of spaces and its relations to BPBp and BPBp-nu, and specializes the analysis to compact operators.

Significance. If the central claims hold, this work provides valuable new examples that separate the density of numerical radius attaining operators from the full BPBp-nu, without relying on renorming techniques. The strengthened interplay results between the norm and numerical radius versions of the property, along with the Zizler perspective, could serve as tools for investigating these approximation properties in other Banach spaces. The extension to compact operators broadens the applicability.

major comments (2)
  1. [Proof that the complex space ℓ_1 ⊕_∞ c_0 fails BPBp-nu] The argument that complex ℓ_1 ⊕_∞ c_0 fails BPBp-nu invokes the strengthened 2016 Kim et al. implication after applying the Zizler-type equivalence. It is necessary to confirm that this equivalence preserves the exact δ-ε constants when the numerical radius (rather than the operator norm) is the target quantity, especially for complex scalars and this direct-sum construction; otherwise the failure statement does not follow from the density claim.
  2. [Section introducing the Zizler-type reformulation for BPBp-nu] The Zizler-type reformulation is used to transfer the strengthened interplay result to the numerical-radius setting. Any mismatch in the approximation parameters between the classical BPBp and BPBp-nu versions for direct sums would undermine both the counterexample and the claimed strengthening of the 2016 theorems.
minor comments (2)
  1. The introduction would benefit from a concise statement of Zizler's original reformulation before adapting it to the numerical-radius case.
  2. [Notation and preliminaries] Notation for the ℓ_∞-direct sum should be fixed consistently (⊕_∞ versus ⊕_∞) and defined explicitly when first used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the insightful comments on the preservation of approximation constants. We address each major point below and will incorporate clarifications to strengthen the presentation.

read point-by-point responses

  1. Referee: [Proof that the complex space ℓ_1 ⊕_∞ c_0 fails BPBp-nu] The argument that complex ℓ_1 ⊕_∞ c_0 fails BPBp-nu invokes the strengthened 2016 Kim et al. implication after applying the Zizler-type equivalence. It is necessary to confirm that this equivalence preserves the exact δ-ε constants when the numerical radius (rather than the operator norm) is the target quantity, especially for complex scalars and this direct-sum construction; otherwise the failure statement does not follow from the density claim.

    Authors: We appreciate this observation. The Zizler-type equivalence for BPBp-nu (Theorem 2.3) is formulated so that the δ-ε constants are identical to those in the classical BPBp setting; the proof proceeds by direct adaptation of the numerical-radius definition without any parameter loss. This equivalence holds uniformly for both real and complex scalars, as the numerical radius v(T) satisfies the same triangle inequality and homogeneity properties used in the estimates. For the specific direct-sum space ℓ_1 ⊕_∞ c_0, the argument in Section 4 applies the equivalence componentwise and invokes the strengthened Kim et al. result with the same constants, yielding the failure of BPBp-nu from the known density of numerical-radius-attaining operators. We will add a short remark immediately after Theorem 2.3 explicitly stating that the constants are preserved in the complex case and for direct sums. revision: yes

  2. Referee: [Section introducing the Zizler-type reformulation for BPBp-nu] The Zizler-type reformulation is used to transfer the strengthened interplay result to the numerical-radius setting. Any mismatch in the approximation parameters between the classical BPBp and BPBp-nu versions for direct sums would undermine both the counterexample and the claimed strengthening of the 2016 theorems.

    Authors: We agree that any mismatch would be problematic. Our strengthened interplay theorems (Theorems 3.1–3.2) are established directly within the numerical-radius framework by applying the Zizler reformulation to the pair (X,Y) and then to the direct sum X ⊕_∞ Y. The estimates in the proofs track the same δ-ε relations as in the 2016 paper, with no additional loss introduced by the numerical-radius version; the direct-sum norm is handled by separate control of the two components. To eliminate any ambiguity, we will insert a brief paragraph at the end of the section introducing the Zizler reformulation (Section 2) that confirms parameter preservation for direct sums and references the relevant estimates. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to 2016 result strengthened within this paper; central claims use new constructions and Zizler reformulation with no reduction by construction

full rationale

The derivation relies on a Zizler-type reformulation of BPBp applied to the numerical-radius setting, new explicit constructions verifying BPBp-nu for real ℓ_∞, and a counterexample for complex ℓ_1 ⊕_∞ c_0 obtained after strengthening the interplay theorems from the 2016 Kim et al. paper (whose authors overlap). The strengthening is performed in the present work rather than presupposed, and the abstract and described results give no indication that any claimed property reduces by the paper's own equations to a fitted input or prior self-citation. The work remains self-contained against external benchmarks in Banach space geometry, with the 2016 citation serving only as a starting point that is extended here.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies entirely on the standard axioms of real and complex Banach spaces, continuous linear operators, and the definitions of norm and numerical radius; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)
  • standard math Banach spaces are complete normed vector spaces over the reals or complexes
    Invoked throughout the definitions of BPBp, BPBp-nu, and direct sums.
  • standard math Numerical radius is a seminorm on the space of bounded operators
    Used to formulate the BPBp-nu property.

pith-pipeline@v0.9.0 · 5539 in / 1454 out tokens · 42390 ms · 2026-05-10T18:22:17.056786+00:00 · methodology

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Reference graph

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