arxiv: 2605.09354 · v1 · submitted 2026-05-10 · 🧮 math.OA · math.FA

Recognition: 2 theorem links

· Lean Theorem

On the spectral radius of operator tuples

Eli Shamovich, Marcel Scherer, Orr Shalit

Pith reviewed 2026-05-12 01:51 UTC · model grok-4.3

classification 🧮 math.OA math.FA

keywords spectral radiusoperator tuplesjoint spectrumoperator spacesquantizationsnoncommutative functionscommuting operators

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The pith

For commuting operator tuples, the spectral radius is determined only by the underlying normed space.

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

This paper shows that for any operator space structure E on a normed space V, and for any commuting tuple of operators X, the associated spectral radius ρ_E(X) is equal to the largest V-norm of a point in the joint spectrum of X. This means the choice of how to equip V with an operator space structure does not affect the spectral radius when the operators commute. A sympathetic reader cares because this reduces a potentially complicated operator-space computation to a classical one involving only the norm and the joint spectrum, while also providing contrasts for non-commuting cases and uniqueness results for the operator spaces themselves.

Core claim

The paper claims that for a commuting operator tuple X and any quantization E of the normed space V, ρ_E(X) = max{‖λ‖_V : λ ∈ σ(X)}, with σ(X) the joint spectrum. This holds independently of the specific operator space structure E. For selfadjoint operator spaces, the spectral radius function determines the space uniquely. The same spectral radius also implies that the algebras of locally uniformly bounded NC functions on the NC unit balls are the same. A key tool is the characterization of ρ_E(A) using the invertibility domain of the associated linear pencil.

What carries the argument

The spectral radius ρ_E defined via the operator space E, which reduces to the max V-norm on the joint spectrum for commuting tuples.

If this is right

For commuting X, ρ_E(X) equals max{‖λ‖_V : λ in σ(X)} for any quantization E of V.
For dim V ≥ 3 there exist matrix tuples X with ρ_min(V)(X) ≠ ρ_max(V)(X).
If two selfadjoint operator spaces have ρ_E1(X) = ρ_E2(X) for all X then E1 = E2.
If two operator spaces induce the same spectral radius then their algebras of locally uniformly bounded NC functions coincide.

Where Pith is reading between the lines

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

Commutativity allows bypassing operator space specifics in spectral calculations, reducing to classical geometry of the normed space.
The linear pencil characterization offers a potential computational tool for spectral radii beyond the commuting case.
Uniqueness of the operator space from the radius function may extend to other classes of spaces or invariants.
This independence might apply to related concepts like the joint spectral radius or other multivariable spectra.

Load-bearing premise

The tuple of operators commutes and the operator space E is a quantization compatible with the norm on V.

What would settle it

A commuting tuple X on C^3 and two different quantizations E1 and E2 of the same V such that ρ_E1(X) differs from ρ_E2(X) would disprove the claim.

read the original abstract

In recent work, Shalit and Shamovich associated to every operator space structure $\mathcal{E}$ on $\mathbb{C}^d$ a spectral radius function $\rho_{\mathcal{E}}$ on $d$-tuples of operators. The main goal of this paper is to elucidate how this spectral radius depends on the operator space structure. Let $V = (\mathbb{C}^d, \|\cdot\|_V)$ be a normed space and let $\mathcal{E}$ be a quantization of $V$. We show that for a commuting operator tuple $X$, the spectral radius depends only on the underlying normed space; more precisely, \[ \rho_{\mathcal{E}}(X) = \max\{ \|\lambda\|_V : \lambda \in \sigma(X)\}, \] where $\sigma(X)$ denotes the joint spectrum of $X$. In contrast, we prove that if $\dim V \geq 3$, then $\rho_{\min(V)}(X) \neq \rho_{\max(V)}(X)$ already for some matrix tuple $X$. When $\mathcal{E}_1$ and $\mathcal{E}_2$ are selfadjoint operator spaces, we show that $\rho_{\mathcal{E}_1}(X) = \rho_{\mathcal{E}_2}(X)$ for all tuples $X$ implies $\mathcal{E}_1 = \mathcal{E}_2$. We present two proofs of this result; a key ingredient in one of them is a characterization, of independent interest, of $\rho_{\mathcal{E}}(A)$ in terms of the invertibility domain of the linear pencil associated with $A$. Finally, we prove that if two operator spaces give rise to the same spectral radius function, then the algebras of locally uniformly bounded NC functions on the corresponding NC unit balls coincide.

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

Commutativity makes the spectral radius depend only on the norm, with non-commuting cases distinguishing structures via a concrete counterexample.

read the letter

For commuting operator tuples the spectral radius is determined solely by the underlying normed space, independent of the operator space quantization. The paper also shows that this independence fails without commutativity, and that matching radii force matching self-adjoint structures and matching noncommutative function algebras. What stands out as new is the explicit formula tying the radius to the joint spectrum for commuting cases, the matrix counterexample separating min and max quantizations in dimension three and higher, the uniqueness result for self-adjoint operator spaces, and the characterization of the radius through the invertibility domain of linear pencils. The pencil tool is presented as interesting in its own right, and the two proofs for uniqueness add some robustness. The work is careful in separating the commuting and non-commuting regimes and in deriving the algebra coincidence from the radius equality. The counterexample provides concrete evidence that the distinction is real rather than formal. One soft spot is that the paper works entirely within finite-dimensional base spaces, so the results do not immediately extend to infinite-dimensional settings where operator spaces behave differently. The step connecting equal radii to identical noncommutative function algebras is stated clearly but would benefit from explicit verification in the text to confirm there are no hidden assumptions. This paper is for specialists in operator theory who care about how operator space structures interact with spectral invariants. Readers already familiar with noncommutative function algebras or multivariable spectral theory will get the most out of the distinctions and the new characterization. It is solid enough on its own terms to merit a serious referee rather than a desk rejection. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper studies the spectral radius ρ_E associated to an operator space structure E on a normed space V = ℂ^d. For any quantization E of V and any commuting d-tuple X, it proves ρ_E(X) = max{‖λ‖_V : λ ∈ σ(X)}, where σ(X) is the joint spectrum. It shows this fails for non-commuting matrix tuples when dim V ≥ 3 by comparing min and max quantizations. For self-adjoint operator spaces, ρ_{E1}(X) = ρ_{E2}(X) for all X implies E1 = E2, with two proofs; one relies on a characterization of ρ_E(A) via the invertibility domain of the associated linear pencil. Finally, equal spectral radii imply that the algebras of locally uniformly bounded NC functions on the corresponding NC unit balls coincide.

Significance. If the central claims hold, the work shows that the spectral radius for commuting tuples depends only on the underlying normed space rather than the full operator-space structure, reducing a noncommutative quantity to a classical one and simplifying its computation. The pencil-invertibility characterization is of independent interest and supports one of the two proofs. The consequence for NC function algebras is a clean structural implication. The explicit counterexample for non-commuting tuples when d ≥ 3 underscores the necessity of commutativity and provides a useful contrast.

minor comments (3)

[Abstract] The abstract states that two proofs are given and that one uses a pencil characterization of independent interest; the introduction or §1 should explicitly cross-reference the theorems containing each proof and the precise statement of the characterization (e.g., the invertibility-domain description of ρ_E(A)).
The definition of a 'quantization E of V' (i.e., an operator-space structure compatible with the given norm on V) should be recalled with a standard reference (e.g., to Pisier or Effros-Ruan) at the first appearance in the introduction, to make the setup self-contained for readers outside operator-space theory.
In the final result on NC function algebras, clarify whether the NC unit balls are taken with respect to the operator-space norms or the underlying vector-space norms; a brief sentence relating the two would prevent ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and constructive report, which highlights the main contributions of the paper and recommends minor revision. We appreciate the recognition of the results on commuting tuples, the counterexamples for non-commuting cases, the pencil characterization, and the implications for NC function algebras. Since no specific major comments were raised, we will proceed with minor revisions to improve clarity and address any typographical or presentation issues in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from definitions

full rationale

The paper introduces ρ_E via prior independent work by two of the authors, then proves that for any quantization E of a normed space V and commuting tuple X, ρ_E(X) equals max{‖λ‖_V : λ ∈ σ(X)}. This equality is derived as a theorem using the joint spectrum and commutativity (with an explicit counter-example showing failure without commutativity), plus an independent characterization of ρ_E via the invertibility domain of the associated linear pencil. The uniqueness result (same ρ implies E1 = E2 for selfadjoint spaces) and the NC-function algebra coincidence are likewise proved from these elements rather than by redefinition or fitted inputs. The single self-citation serves only to recall the definition of ρ_E and carries no load-bearing circularity for the new claims.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the standard definition of an operator-space quantization of a normed space, the definition of the joint spectrum for commuting tuples, and the definition of the linear pencil associated to a matrix tuple. No free parameters or newly invented entities appear in the abstract.

axioms (2)

domain assumption E is a quantization of the normed space V (i.e., the matrix norms on E are compatible with the given norm on V)
Invoked when stating that ρ_E depends only on V for commuting tuples.
standard math The joint spectrum σ(X) is well-defined for commuting operator tuples
Used directly in the equality ρ_E(X) = max ‖λ‖_V.

pith-pipeline@v0.9.0 · 5625 in / 1616 out tokens · 50321 ms · 2026-05-12T01:51:15.661406+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
ρ_E(X) = max{‖λ‖_V : λ ∈ σ(X)} for commuting X; ρ_min(V)(X) ≠ ρ_max(V)(X) already for some matrix tuple when dim V ≥ 3
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
characterization of ρ_E(A) in terms of the invertibility domain of the linear pencil L_A

Reference graph

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