Joint Numerical Radius of Tuples

• Joint numerical radius is defined as the p-norm of the numerical values of an operator n-tuple, generalizing the classical numerical radius.
• It satisfies key properties such as homogeneity, triangle inequality, and unitary invariance, which underpin sharp norm inequalities and spectral estimates.
• Extensions to functional calculus and matrix convexity enable advanced applications in spectral analysis, semigroup stability, and convex optimization.

The joint numerical radius of tuples generalizes the classical numerical radius of a single operator by measuring the “size” of an $n$-tuple of bounded linear operators on a Hilbert or Banach space through a vector $p$-norm of their numerical values. This construction, and its variants in several contexts (Hilbert spaces, semi-Hilbert spaces, Banach spaces), serve as a central tool in operator theory, providing sharp norm inequalities, spectral and convexity properties, and subdifferential calculus applicable to matrix analysis, multipartite operator inequalities, and convex optimization.

1. Definitions and Foundational Examples

Given a complex Hilbert space $\mathcal{H}$ and the algebra $\mathbb{B}(\mathcal{H})$ of bounded linear operators, the classical numerical radius of $A \in \mathbb{B}(\mathcal{H})$ is $w(A) = \sup\{|\langle Ax, x \rangle|:\|x\|=1\}$.

The joint $p$-numerical radius of an $n$-tuple $(T_1,\dots,T_n)\in\mathbb{B}(\mathcal{H})^n$ is defined for $p \ge 1$ as

$p$0

• For $p$1, this reduces to the numerical radius $p$2.
• For $p$3, $p$4 coincides with the Euclidean operator radius (Popescu) (Moslehian et al., 2015), and is sometimes denoted $p$5.
• Variants with $p$6 yield the classical joint numerical radius (sum-norm on the tuple's numerical values) (Feki et al., 2024).
• For Banach spaces $p$7, the generalization involves pairs $p$8 with $p$9, $\mathcal{H}$0, $\mathcal{H}$1 (Mal, 2022).

2. Basic Properties and Inequalities

The mapping $\mathcal{H}$2 defines a norm on $\mathcal{H}$3. Principal properties include:

• Homogeneity: $\mathcal{H}$4.
• Triangle inequality: $\mathcal{H}$5 (by Minkowski inequality in $\mathcal{H}$6) (Moslehian et al., 2015, Feki et al., 2024).
• Unitary invariance: $\mathcal{H}$7 for unitary $\mathcal{H}$8.
• Adjoint symmetry: $\mathcal{H}$9.
• Norm domination: $\mathbb{B}(\mathcal{H})$0.
• Two-sided bounds: For $\mathbb{B}(\mathcal{H})$1 (Euclidean radius), $\mathbb{B}(\mathcal{H})$2 (Feki et al., 2024, Bhunia et al., 2023); in general, $\mathbb{B}(\mathcal{H})$3 for $\mathbb{B}(\mathcal{H})$4 (Feki et al., 2024).

Sharp inequalities interpolate between the numerical radius ($\mathbb{B}(\mathcal{H})$5), the Euclidean radius ($\mathbb{B}(\mathcal{H})$6), and higher moment-type bounds for $\mathbb{B}(\mathcal{H})$7. Comparison formulas:

• For $\mathbb{B}(\mathcal{H})$8, $\mathbb{B}(\mathcal{H})$9 (Moslehian et al., 2015).
• For $A \in \mathbb{B}(\mathcal{H})$0, $A \in \mathbb{B}(\mathcal{H})$1 (Moslehian et al., 2015).
• Bounds via block structure: $A \in \mathbb{B}(\mathcal{H})$2 (Moslehian et al., 2015).

3. Functional Calculus and Refinements

The joint numerical radius admits refinements based on operator functional calculus. The main result (Moslehian et al., 2015): If $A \in \mathbb{B}(\mathcal{H})$3 are continuous and satisfy $A \in \mathbb{B}(\mathcal{H})$4, then for any $A \in \mathbb{B}(\mathcal{H})$5, $A \in \mathbb{B}(\mathcal{H})$6, $A \in \mathbb{B}(\mathcal{H})$7,

$A \in \mathbb{B}(\mathcal{H})$8

Specializing $A \in \mathbb{B}(\mathcal{H})$9 and $w(A) = \sup\{|\langle Ax, x \rangle|:\|x\|=1\}$0 gives hybrid power-type estimates: $w(A) = \sup\{|\langle Ax, x \rangle|:\|x\|=1\}$1 In the Euclidean case $w(A) = \sup\{|\langle Ax, x \rangle|:\|x\|=1\}$2, this recovers and generalizes Dragomir’s two-operator inequalities (Moslehian et al., 2015, Bhunia et al., 4 Mar 2026, Bhunia et al., 2023).

4. Variants: Semi-Hilbert Spaces and the $w(A) = \sup\{|\langle Ax, x \rangle|:\|x\|=1\}$3-Joint Numerical Radius

For a positive semi-definite operator $w(A) = \sup\{|\langle Ax, x \rangle|:\|x\|=1\}$4 on $w(A) = \sup\{|\langle Ax, x \rangle|:\|x\|=1\}$5, the semi-inner-product is $w(A) = \sup\{|\langle Ax, x \rangle|:\|x\|=1\}$6, with the $w(A) = \sup\{|\langle Ax, x \rangle|:\|x\|=1\}$7-seminorm $w(A) = \sup\{|\langle Ax, x \rangle|:\|x\|=1\}$8. The $w(A) = \sup\{|\langle Ax, x \rangle|:\|x\|=1\}$9-joint numerical radius of $p$0 is

$p$1

This seminorm generalizes $p$2 and satisfies analogous bounds and interpolation formulas (Bhunia et al., 30 Jun 2025): $p$3 If $p$4 is $p$5-normal and commuting, then equality occurs (Bhunia et al., 30 Jun 2025).

Extensions to $p$6-numerical radius inequalities for two tuples (Feki, 2020, Jana et al., 2023) yield refined parallelogram, Buzano-type, and convex-combination bounds, and connect to the $p$7-Davis–Wielandt radius.

5. Convexity, Extreme Points, and Subdifferential Characterization

The joint numerical radius $p$8 is convex in the tuple: $p$9 (Grover et al., 2022, Mal, 2022, Mal, 7 Jul 2025). In finite-dimensional settings, the maximal value in the definition is attained at an extreme point of the underlying unit sphere (Choquet boundary) (Mal, 2022, Mal, 7 Jul 2025).

The subdifferential of $n$0 admits an explicit convex hull formula. For $n$1,

$n$2

where $n$3 is the set of maximizers in the definition (Grover et al., 2022). The Gâteaux derivative and smoothness behavior of $n$4 on Banach spaces are described by convex-analytic formulas in terms of these maximizers and their supporting functionals (Mal, 7 Jul 2025).

6. Matrix Convexity, Toeplitz Contractivity, and Operator Systems

A nonclassical development is the matrix convexity theory of the joint numerical radius for operator tuples (Farenick, 2024). Given $n$5-tuples $n$6 on a Hilbert space, the minimal and maximal matrix convex hulls of the joint numerical range correspond to sets where the generalized joint numerical radius $n$7 and the Toeplitz modulus $n$8, defined via positivity of an associated Toeplitz matrix, do not exceed 1. For $n$9 one recovers the classical $(T_1,\dots,T_n)\in\mathbb{B}(\mathcal{H})^n$0. For $(T_1,\dots,T_n)\in\mathbb{B}(\mathcal{H})^n$1, the scaling constant $(T_1,\dots,T_n)\in\mathbb{B}(\mathcal{H})^n$2 for the inclusion $(T_1,\dots,T_n)\in\mathbb{B}(\mathcal{H})^n$3 is at least 2, but the optimal value remains unresolved.

A Toeplitz-contractive $(T_1,\dots,T_n)\in\mathbb{B}(\mathcal{H})^n$4-tuple corresponds to the existence of a unitary dilation (generalizing the Halmos theorem), and the sets $(T_1,\dots,T_n)\in\mathbb{B}(\mathcal{H})^n$5 and $(T_1,\dots,T_n)\in\mathbb{B}(\mathcal{H})^n$6 are matrix convex sets.

7. Applications and Further Developments

The joint numerical radius and its generalizations provide a unifying lens for many operator-analytic inequalities:

• Refined operator inequalities: Interpolation between norm and numerical radius (and beyond) improves bounds on commutators, block-matrix operators, and spectral estimates (Bhunia et al., 4 Mar 2026, Bhunia et al., 2023).
• Spectral radius and semigroup stability: Functional calculus bounds derived from $(T_1,\dots,T_n)\in\mathbb{B}(\mathcal{H})^n$7 are used in semigroup theory, sectorial operator stability, and PDE analysis (Ismailov et al., 19 Mar 2026).
• Aluthge transforms: The behavior of the joint numerical radius under multi-variable Aluthge transforms relates to spectral radius formulas and contractivity criteria in tuple dynamics (Feki et al., 2020).
• Banach space theory: The joint numerical index and its lower bounds interpolate structural constants of Banach spaces, with sharp constants computed for classical spaces (Mal, 2022).
• Subdifferential/Optimization: Exact subdifferential characterizations enable best-approximation problems, orthogonality notions, and differentiability analysis in convex and normed operator functionals (Grover et al., 2022, Mal, 7 Jul 2025).
• Quasi-normed and sectorial settings: The introduction of gauge functions and admissibility thresholds $(T_1,\dots,T_n)\in\mathbb{B}(\mathcal{H})^n$8 further refines the joint numerical radius to non-convex and sectorial contexts (Ismailov et al., 19 Mar 2026).

These frameworks collectively position the joint numerical radius as a central object in multivariable operator theory and its applications.