[Paper Review] Unitary invariants in multivariable operator theory

Gelu Popescu|arXiv (Cornell University)|Oct 22, 2004

Holomorphic and Operator Theory36 references4 citations

TL;DR

This paper introduces a multivariable generalization of the classical numerical radius for $n$-tuples of non-commuting operators, establishing it as a unitary invariant and proving its equivalence to the operator norm. The key contribution is a multivariable von Neumann-type inequality and related operator norm estimates involving noncommutative polynomials and Fock space representations, with applications to constrained interpolation and positivity conditions on group $C^*$-algebras.

ABSTRACT

The problems considered in this paper come as a natural continuation of our program to develop a free analogue of Sz.-Nagy-Foias theory, for row contractions. The paper is structured as follows: Introduction Part I. Unitary invariants for n-tuples of operators 1. Joint numerical radius 2. Euclidean operator radius 3. Joint numerical range and spectrum 4. $ρ$-operator radius Part II. Joint operator radii, inequalities, and applications 5. von Neumann inequalities 6. Constrained von Neumann inequalities 7. Multivariable Haagerup-de la Harpe inequalities 8. Multivariable Fejer inequalities References

Motivation & Objective

• To develop a unitary invariant framework for $n$-tuples of non-commuting bounded operators in multivariable operator theory.
• To generalize the classical numerical radius to $n$-tuples using the full Fock space and free semigroups.
• To establish multivariable versions of the von Neumann inequality and related operator norm inequalities.
• To characterize positivity conditions for noncommutative polynomials in terms of operator coefficients and trigonometric bounds.
• To extend classical interpolation and dilation theorems to noncommutative settings using $F_n^\infty$ and $\mathcal{A}_n$ algebras.

Proposed method

• Define the joint numerical radius $w(T_1,\ldots,T_n)$ via a supremum over vector families in the Fock space with unit $\ell^2$-norm.
• Use the left creation operators $S_1,\ldots,S_n$ on the full Fock space $F^2(H_n)$ as noncommutative analogues of the shift operator.
• Establish equivalence between the joint numerical radius and the operator norm using completely positive maps on operator systems generated by $S_1,\ldots,S_n$.
• Prove a multivariable von Neumann inequality: $w(f_1(T),\ldots,f_k(T)) \leq \|[f_1,\ldots,f_k]\| + 2\left(\sum_{j=1}^n |f_j(0)|^2\right)^{1/2}$ for $f_j \in \mathcal{A}_n$.
• Derive trigonometric bounds on coefficients of noncommutative polynomials via positivity in $C^*({\mathbb{F}}_n)$ and $C_{\text{red}}^*({\mathbb{F}}_n)$, using the universal property of $\bf{U}_i$.
• Apply the Berger-Kato-Stampfli mapping theorem and dilation theorems to characterize joint numerical ranges and operator radii in terms of $F_n^\infty$-modules.

Experimental results

Research questions

• RQ1 How can the classical numerical radius be generalized to $n$-tuples of non-commuting operators in a unitary-invariant way?
• RQ2 What is the relationship between the joint numerical radius and the operator norm for $n$-tuples of operators?
• RQ3 Can a multivariable von Neumann inequality be established for noncommutative polynomials in $\mathcal{A}_n$?
• RQ4 What are the coefficient constraints on noncommutative polynomials that ensure positivity in the reduced or full group $C^*$-algebras of the free group $\mathbb{F}_n$?
• RQ5 How do the joint numerical range and spectrum of $n$-tuples relate to the structure of the noncommutative disc algebra $\mathcal{A}_n$?

Key findings

• The joint numerical radius $w(T_1,\ldots,T_n)$ is a norm equivalent to the operator norm on $B(\mathcal{H})^{(n)}$, providing a unitary invariant for $n$-tuples.
• The unit ball $\{(T_1,\ldots,T_n) : w(T_1,\ldots,T_n) \leq 1\}$ is characterized via completely positive maps on operator systems generated by $S_1,\ldots,S_n$.
• A multivariable von Neumann inequality holds: $w(f_1(T),\ldots,f_k(T)) \leq \|[f_1,\ldots,f_k]\| + 2\left(\sum_{j=1}^n |f_j(0)|^2\right)^{1/2}$ for $f_j \in \mathcal{A}_n$.
• For a noncommutative polynomial $f(X_1,\ldots,X_n)$ of degree $m-1$, if $f(S_1,\ldots,S_n) \geq 0$, then $\left(\sum_{|\alpha|=k} |a_\alpha|^2\right)^{1/2} \leq a_0 \cos\left(\frac{\pi}{[\frac{m-1}{k}]+2}\right)$ for $1 \leq k \leq m-1$.
• If $f(U_1,\ldots,U_n) \geq 0$ in $C_{\text{red}}^*({\mathbb{F}}_n)$ or $f(\bf{U}_1,\ldots,\bf{U}_n) \geq 0$ in $C^*({\mathbb{F}}_n)$, the same coefficient bound holds due to the universal property and restriction to $F^2(H_n)$.