[Paper Review] Joint numerical radius of spherical Aluthge transforms of tuples of Hilbert space operators

Kais Feki, Takeaki Yamazaki|arXiv (Cornell University)|Apr 6, 2020

Holomorphic and Operator Theory21 references4 citations

TL;DR

This paper introduces the spherical Aluthge transform for tuples of Hilbert space operators and establishes sharp inequalities relating the joint numerical radius and joint operator norm of the transformed tuple. It proves that the joint spectral radius of a tuple equals the limit of the joint operator norm of its iterated spherical Aluthge transforms, providing a new spectral characterization via iterative transformation.

ABSTRACT

Let $\mathbf{T}=(T_1,\ldots,T_d)$ be a $d$-tuple of operators on a complex Hilbert space $\mathcal{H}$. The spherical Aluthge transform of $\mathbf{T}$ is the $d$-tuple given by $\widehat{\mathbf{T}}:=(\sqrt{P}V_1\sqrt{P},\ldots,\sqrt{P}V_d\sqrt{P})$ where $P:=\sqrt{T_1^*T_1+\ldots+T_d^*T_d}$ and $(V_1,\ldots,V_d)$ is a joint partial isometry such that $T_k=V_k P$ for all $1 \le k \le d$. In this paper, we prove several inequalities involving the joint numerical radius and the joint operator norm of $\widehat{\mathbf{T}}$. Moreover, a characterization of the joint spectral radius of an operator tuple $\mathbf{T}$ via $n$-th iterated of spherical Aluthge transform is established.

Motivation & Objective

• To define and analyze the spherical Aluthge transform for d-tuples of Hilbert space operators.
• To derive sharp inequalities between the joint numerical radius and joint operator norm of the spherical Aluthge transform.
• To characterize the joint spectral radius of an operator tuple using the limit of the joint operator norms of its iterated spherical Aluthge transforms.

Proposed method

• The spherical Aluthge transform of a d-tuple T = (T₁,…,Tₙ) is defined as (P¹ᐟ²V₁P¹ᐟ²,…,P¹ᐟ²VₙP¹ᐟ²), where P = (ΣTₖ* Tₖ)¹ᐟ² and Tₖ = VₖP via joint polar decomposition.
• The joint numerical radius ω(T) is defined as the supremum of the ℓ²-norm of the joint numerical range over the unit sphere.
• The joint operator norm ‖T‖ is defined as the supremum of (∑‖Tₖx‖²)¹ᐟ² over the unit sphere.
• The paper uses iterative application of the spherical Aluthge transform and analyzes the asymptotic behavior of the joint operator norms of the iterates.
• Key inequalities are derived using properties of the polar decomposition and submultiplicativity of operator norms.
• A limiting argument based on the convergence of the joint operator norm of iterated transforms is used to characterize the joint spectral radius.

Experimental results

Research questions

• RQ1 How does the joint numerical radius of the spherical Aluthge transform relate to that of the original operator tuple?
• RQ2 What is the relationship between the joint operator norm of the spherical Aluthge transform and the joint operator norm of the original tuple?
• RQ3 Can the joint spectral radius of a d-tuple of operators be characterized through the iterated spherical Aluthge transform?
• RQ4 Does the joint operator norm of the iterated spherical Aluthge transforms converge to the joint spectral radius?
• RQ5 What are the sharp bounds for the joint numerical radius in terms of the joint operator norm for the spherical Aluthge transform?

Key findings

• The joint numerical radius of the spherical Aluthge transform satisfies ω(Ť) ≤ ω(T), establishing a monotonicity property under the transform.
• The joint operator norm of the spherical Aluthge transform satisfies ‖Ť‖ ≤ ‖T‖, showing norm non-increase.
• The joint spectral radius r(T) is equal to the limit of the joint operator norm of the n-th iterated spherical Aluthge transform, i.e., r(T) = limₙ→∞ ‖Ťₙ‖.
• For any k ≥ 1, the joint operator norm of the k-th power of the iterated transform satisfies ‖Ťₙᵏ‖ ≤ ‖Tᵏ⁺¹‖¹ᐟ²‖Tᵏ⁻¹‖¹ᐟ².
• The joint operator norm of the iterated spherical Aluthge transforms converges to the joint spectral radius s, and this limit is shown to equal r(T) via contradiction.
• The joint spectral radius is characterized as the limit of the joint operator norm of the iterated spherical Aluthge transforms, providing a new spectral representation.