Numerical radius and $\ell_p$ operator norm of Kronecker products and Schur powers: inequalities and equalities

TL;DR

This paper investigates inequalities involving the numerical radius and operator norms of Kronecker products, extending classical bounds, characterizing equality cases, and applying results to Schur products and polynomial root estimations.

Contribution

It refines bounds on the numerical radius of Kronecker products, characterizes when equality holds, and extends results to semi-Hilbertian spaces and Schur products, also improving polynomial root estimates.

Findings

Refined inequality for numerical radius of Kronecker products.

Characterization of when $w(A ensor B) = w(A) orm{B}$.

Derived bounds for $ ext{ell}_p$ operator norm and numerical radius, with applications to polynomial roots.

Abstract

Suppose $A=[a_{ij}]\in \mathcal{M}_n(\mathbb{C})$ is a complex $n \times n$ matrix and $B\in \mathcal{B}(\mathcal{H})$ is a bounded linear operator on a complex Hilbert space $\mathcal{H}$. We show that $w(A\otimes B)\leq w(C),$ where $w(\cdot)$ denotes the numerical radius and $C=[c_{ij}]$ with $c_{ij}= w\left(\begin{bmatrix} 0& a_{ij}\\ a_{ji}&0 \end{bmatrix} \otimes B\right).$ This refines Holbrook's classical bound $w(A\otimes B)\leq w(A) \|B\|$ [J. Reine Angew. Math. 1969], when all entries of $A$ are non-negative. If moreover $a_{ii}\neq 0$ $ \forall i$, we prove that $w(A\otimes B)= w(A) \|B\|$ if and only if $w(B)=\|B\|.$ We then extend these and other results to the more general setting of semi-Hilbertian spaces induced by a positive operator. In the reverse direction, we also specialize these results to Kronecker products and hence to Schur/entrywise products, of matrices: (1)(a) We first provide an alternate proof (using $w(A)$) of a result of Goldberg-Zwas [Linear Algebra Appl. 1974] that if the spectral norm of $A$ equals its spectral radius, then each Jordan block for each maximum-modulus eigenvalue must be $1 \times 1$ ("partial diagonalizability"). (b) Using our approach, we further show given $m \geq 1$ that $w(A^{\circ m})\leq w^m(A)$ - we also characterize when equality holds here. (2) We provide upper and lower bounds for the $\ell_p$ operator norm and the numerical radius of $A\otimes B$ for all $A \in \mathcal{M}_n(\mathbb{C})$, which become equal when restricted to doubly stochastic matrices $A$. Finally, using these bounds we obtain an improved estimation for the roots of an arbitrary complex polynomial.