---
title: Joint Numerical Radius of Tuples
url: https://www.emergentmind.com/topics/joint-numerical-radius-of-tuples
type: topic
---

# Joint Numerical Radius of Tuples

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
\[
w_p(T_1,\dots,T_n) = \sup_{\|x\|=1}\left(\sum_{i=1}^n |\langle T_i x, x\rangle|^p\right)^{1/p}.
\]
– For $n=1$, this reduces to the numerical radius $w(T)$.  
– For $p=2$, $w_2$ coincides with the Euclidean operator radius (Popescu) [1502.00083], and is sometimes denoted $w_e$.  
– Variants with $p=1$ yield the classical joint numerical radius (sum-norm on the tuple's numerical values) [2410.03669].  
– For Banach spaces $X$, the generalization involves pairs $(x, x^*)$ with $x^* \in X^*$, $\|x\| = \|x^*\| = 1$, $x^*(x) = 1$ [2212.06500].

## 2. Basic Properties and Inequalities

The mapping $(T_1,\dots,T_n)\mapsto w_p(T_1,\dots,T_n)$ defines a norm on $\mathbb{B}(\mathcal{H})^n$. Principal properties include:
- **Homogeneity:** $w_p(\lambda T_1, \dots, \lambda T_n) = |\lambda| w_p(T_1,\dots,T_n)$.
- **Triangle inequality:** $w_p(T_1+S_1, \dots, T_n+S_n) \le w_p(T_1, \dots, T_n) + w_p(S_1, \dots, S_n)$ (by Minkowski inequality in $\mathbb{C}^n$) [1502.00083, 2410.03669].
- **Unitary invariance:** $w_p(U^*T_1U,\dots,U^*T_nU) = w_p(T_1,\dots,T_n)$ for unitary $U$.
- **Adjoint symmetry:** $w_p(T_1^*,\dots,T_n^*) = w_p(T_1,\dots,T_n)$.
- **Norm domination:** $w_p(T_1,\dots,T_n)\le \sum_{i=1}^n\|T_i\|$.
- **Two-sided bounds:** For $p=2$ (Euclidean radius), $\frac{1}{\sqrt{n}}\|\mathbf{T}\| \le w_2(\mathbf{T}) \le \|\mathbf{T}\|$ [2410.03669, 2308.09258]; in general, $|p| \|\cdot\| \le w_p(\cdot)\le \|\cdot\|$ for $p\le1$ [2410.03669].

Sharp inequalities interpolate between the numerical radius ($p=1$), the Euclidean radius ($p=2$), and higher moment-type bounds for $p>2$. Comparison formulas:
- For $p\ge q\ge1$, $w_p(B,C)\le w_q(B,C)\le 2^{1/p-1/q} w_p(B,C)$ [1502.00083].
- For $p\ge2$, $w_p(B,C)\ge 2^{(2-p)/p}\|B^*B+C^*C\|^{1/2}$ [1502.00083].
- Bounds via block structure: $w_p(B,C)\ge 2^{1/p-1} \max\{w(B+C), w(B-C)\}$ [1502.00083].

## 3. Functional Calculus and Refinements

The joint numerical radius admits refinements based on operator functional calculus. The main result [1502.00083]:
If $f,g:[0,\infty)\to [0,\infty)$ are continuous and satisfy $f(t)g(t)=t$, then for any $A_i,B_i,T_i\in\mathbb{B}(\mathcal{H})$, $p\ge 1$, $r\ge1$,
\[
w_p^{rp}(A_1^*T_1B_1, \ldots, A_n^*T_nB_n)
\le \tfrac12\left\| \sum_{i=1}^n \left[B_i^*f^2(|T_i|)B_i\right]^{rp} + \left[A_i^*g^2(|T_i^*|)A_i\right]^{rp} \right\|.
\]
Specializing $A_i = B_i = I$ and $f(t) = t^a, g(t) = t^{1-a}$ gives hybrid power-type estimates:
\[
w_p^p(T_1, ..., T_n) \le \tfrac12\left\| \sum_{i=1}^n |T_i|^{2ap} + |T_i^*|^{2(1-a)p} \right\|.
\]
In the Euclidean case $p=2$, this recovers and generalizes Dragomir’s two-operator inequalities [1502.00083, 2603.03962, 2308.09258].

## 4. Variants: Semi-Hilbert Spaces and the $A$-Joint Numerical Radius

For a positive semi-definite operator $A$ on $\mathcal{H}$, the semi-inner-product is $\langle x, y\rangle_A = \langle Ax, y\rangle$, with the $A$-seminorm $\|x\|_A = \langle Ax,x \rangle^{1/2}$. The $A$-joint numerical radius of $(T_1,\dots,T_n)$ is
\[
\omega_A(T_1, \dots, T_n) = \sup_{\|x\|_A=1}\left( \sum_{k=1}^n |\langle T_k x, x\rangle_A|^2\right)^{1/2}.
\]
This seminorm generalizes $w_2$ and satisfies analogous bounds and interpolation formulas [2506.23642]:
\[
\frac{1}{2\sqrt{n}}\|\mathbf{T}\|_A \le \omega_A(\mathbf{T}) \le \|\mathbf{T}\|_A.
\]
If $(T_1,\dots,T_n)$ is $A$-normal and commuting, then equality occurs [2506.23642].

Extensions to $A$-numerical radius inequalities for two tuples [2005.04758, 2308.09261] yield refined parallelogram, Buzano-type, and convex-combination bounds, and connect to the $A$-Davis–Wielandt radius.

## 5. Convexity, Extreme Points, and Subdifferential Characterization

The joint numerical radius $w_p$ is convex in the tuple: $w_p(S+T)\le w_p(S)+w_p(T)$ [2201.03055, 2212.06500, 2507.04700]. In finite-dimensional settings, the maximal value in the definition is attained at an extreme point of the underlying unit sphere (Choquet boundary) [2212.06500, 2507.04700].

The subdifferential of $w_p$ admits an explicit convex hull formula. For $A = (A_1, ..., A_d) \in M_n(\mathbb{C})^d$,
\[
\partial w(A) = \mathrm{conv}\left\{ \left(\tfrac{\langle x, A_1x \rangle}{w(A)} xx^*, ..., \tfrac{\langle x,A_dx\rangle}{w(A)} xx^* \right) : x \in \Sigma(A) \right\},
\]
where $\Sigma(A)$ is the set of maximizers in the definition [2201.03055]. The Gâteaux derivative and smoothness behavior of $w_p$ on Banach spaces are described by convex-analytic formulas in terms of these maximizers and their supporting functionals [2507.04700].

## 6. Matrix Convexity, Toeplitz Contractivity, and Operator Systems

A nonclassical development is the matrix convexity theory of the joint numerical radius for operator tuples [2408.11011]. Given $d$-tuples $T$ 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 $w(T)$ and the Toeplitz modulus $p(T)$, defined via positivity of an associated Toeplitz matrix, do not exceed 1. For $d=1$ one recovers the classical $w(T)\le \|T\| \le 2w(T)$. For $d>1$, the scaling constant $c_d$ for the inclusion $w(T)\le1 \implies p(T)\le c_d$ is at least 2, but the optimal value remains unresolved.

A Toeplitz-contractive $d$-tuple corresponds to the existence of a unitary dilation (generalizing the Halmos theorem), and the sets $\{T: w(T)\le1\}$ and $\{T: p(T)\le1\}$ 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 [2603.03962, 2308.09258].
- **Spectral radius and semigroup stability:** Functional calculus bounds derived from $w_p$ are used in semigroup theory, sectorial operator stability, and PDE analysis [2603.18405].
- **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 [2004.02538].
- **Banach space theory:** The joint numerical index and its lower bounds interpolate structural constants of Banach spaces, with sharp constants computed for classical spaces [2212.06500].
- **Subdifferential/Optimization:** Exact subdifferential characterizations enable best-approximation problems, orthogonality notions, and differentiability analysis in convex and normed operator functionals [2201.03055, 2507.04700].
- **Quasi-normed and sectorial settings:** The introduction of gauge functions and admissibility thresholds $(f,\delta)$ further refines the joint numerical radius to non-convex and sectorial contexts [2603.18405].

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

Source: https://www.emergentmind.com/topics/joint-numerical-radius-of-tuples