On the spectral radius of operator tuples

Abstract: In recent work, Shalit and Shamovich associated to every operator space structure $\mathcal{E}$ on $\mathbb{C}^d$ a spectral radius function $ρ{\mathcal{E}}$ 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, [ ρ{\mathcal{E}}(X) = \max{ |λ|V : λ\in σ(X)}, ] where $σ(X)$ denotes the joint spectrum of $X$. In contrast, we prove that if $\dim V \geq 3$, then $ρ{\min(V)}(X) \neq ρ{\max(V)}(X)$ already for some matrix tuple $X$. When $\mathcal{E}_1$ and $\mathcal{E}_2$ are selfadjoint operator spaces, we show that $ρ{\mathcal{E}1}(X) = ρ{\mathcal{E}2}(X)$ 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 $ρ{\mathcal{E}}(A)$ 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.

• The paper demonstrates that for commuting tuples, the spectral radius is fully determined by the underlying normed space structure.
• It distinguishes operator space structures by revealing that noncommuting tuples exhibit differing minimal and maximal spectral radii.
• The work connects spectral radius functions with operator algebras by characterizing their invariance, similarity transformations, and analytic domains.

Spectral Radius of Operator Tuples: Dependence on Operator Space Structure

Introduction

This essay provides a detailed survey of "On the spectral radius of operator tuples" (2605.09354), focusing on the interplay between the spectral radius function and operator space structures, especially for tuples of operators in both commutative and noncommutative settings. The paper rigorously establishes the dependence of joint spectral radius functions on the quantization of finite-dimensional normed spaces, clarifies the structural relationships between minimal and maximal operator space quantizations, and explores the consequences of spectral radius equality on analytic noncommutative function algebras.

Spectral Radius Functions Associated With Operator Spaces

The spectral radius for a tuple of operators generalizes classical notions, notably Gelfand's formula, and has broad utility in the spectral theory of Banach algebras, $C^*$-algebras, and operator spaces. Following Shalit and Shamovich [43], every operator space structure $\mathcal{E}$ on $\mathbb{C}^d$ gives rise to a spectral radius function $p_{\mathcal{E}}$ defined on $d$-tuples of operators $X$. Two conceptually distinct spectral radii are defined: the "minimal" spectral radius $p_{\text{min}}$ and the "correct" spectral radius $p_{\mathcal{E}}$ involving the Haagerup tensor product.

The spectral radius function retains invariance under simultaneous similarity transformations. The key result is that for a commuting operator tuple $X$, $p_{\mathcal{E}}(X) < 1$ if and only if $\mathcal{E}$0 is similar to a point in the open operator space ball $\mathcal{E}$1, generalizing the classical similarity orbit characterization.

Characterization of Spectral Radius in the Commutative Setting

A comprehensive characterization is established for commuting tuples:

Theorem A: For $\mathcal{E}$2 and any operator space structure $\mathcal{E}$3 quantizing $\mathcal{E}$4, if $\mathcal{E}$5 is a commuting $\mathcal{E}$6-tuple, then $\mathcal{E}$7 is independent of the quantization:

$\mathcal{E}$8

where $\mathcal{E}$9 denotes the Banach algebra joint spectrum. This implies that the spectral radius for commuting tuples is determined solely by the underlying normed space structure.

The proof leverages the Taylor spectrum and Banach algebraic spectrum equivalence, the functional calculus for analytic functions, and boundedness of relevant homomorphisms. The result ensures that for commuting tuples, the joint spectral radius corresponds precisely to the maximal norm of spectrum points.

Spectral Radius Distinguishing Operator Space Structures

For noncommuting tuples, the spectral radius can differentiate operator space structures:

Theorem B: If $\mathbb{C}^d$0 and $\mathbb{C}^d$1 are distinct selfadjoint operator space structures quantizing the same involutive normed space, then there exists a tuple $\mathbb{C}^d$2 such that $\mathbb{C}^d$3. This claim is substantiated via an explicit construction using matrix tuples and exploiting involutive properties.

The separation between minimal and maximal quantizations is always achievable for spaces of dimension $\mathbb{C}^d$4:

Theorem C: For any complex Banach space $\mathbb{C}^d$5 with $\mathbb{C}^d$6, there exist $\mathbb{C}^d$7 and $\mathbb{C}^d$8 such that $\mathbb{C}^d$9. The proof utilizes Banach-Mazur distance estimates, John's ellipsoid theorem, and representation theory of exterior algebras to obtain strong lower bounds on the separation.

Structural Uniqueness via Spectral Radius Functions

The paper establishes that spectral radius functions encode the operator space structure:

Strong Claim: For selfadjoint operator spaces, the equality $p_{\mathcal{E}}$0 for all tuples $p_{\mathcal{E}}$1 implies $p_{\mathcal{E}}$2. Operator space duality and evaluation at generator tuples are employed to construct two proofs—one via the invertibility domain of the associated linear pencil, another via NC function theory.

This structural rigidity does not necessarily extend to non-selfadjoint operator spaces, as demonstrated in concrete examples. The distinction between the row and column operator space structures, and their associated balls, is analyzed via similarity minimizers and eigenvalue arguments.

Minimal Spectral Radius and Rota-Strang Formulation

The minimal spectral radius is shown to coincide with the Rota-Strang joint spectral radius for certain quantizations, but in general differs from the "correct" spectral radius. For commuting tuples, all spectral radius definitions coincide and admit the formula:

$p_{\mathcal{E}}$3

where $p_{\mathcal{E}}$4 is the product along the word $p_{\mathcal{E}}$5. For $p_{\mathcal{E}}$6-norms, Müller's formula provides an elegant link between spectral radii and index-based growth rates.

Operator Pencils, Domains, and Realizations

The spectral radius is also characterized in terms of the invertibility of linear pencils associated with operator tuples. The spectral radius bounds the largest ball on which the pencil remains invertible, connecting spectral theory with domains of noncommutative analytic functions (NC rational functions and their operator realizations).

A strong characterization (Theorem E) relates $p_{\mathcal{E}}$7 to the domain of realization pencils, the similarity envelope, and analytic function domains. The operator-level generalization is essential, as differences appear for infinite-dimensional operator spaces, illustrated via the full Fock space shifts.

Consequences in Noncommutative Function Theory

If two operator space structures yield identical spectral radius functions, then their algebras of locally uniformly bounded NC analytic functions on the corresponding NC balls coincide, and the identity map is a complete homeomorphism between their function algebras. Conversely, the implication does not hold in general, as counterexamples demonstrate.

The spectral radius functions thus classify NC function algebras up to complete homeomorphism, intertwining spectral invariants with functional analytic structure.

Conclusion

This paper systematically clarifies the dependence of joint spectral radius functions on operator space quantizations. For commuting tuples, the spectral radius is insensitive to quantization; for noncommuting tuples, it distinguishes operator space structures, especially selfadjoint ones. Separation between minimal and maximal quantizations is established for spaces of dimension at least three. These results have concrete implications in NC function theory and operator algebra. Future work may focus on further refining the structural classification of operator spaces via spectral invariants and extending the analysis to infinite-dimensional settings and arbitrary NC domains.