Numerical radius and ${\ell }_p$ operator norm of Kronecker products and Schur powers: inequalities and equalities

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 $w(A\otimes B)\le w(C),$ where $w(\cdot )$ denotes the numerical radius and $C=[c_{ij}]$ with $c_{ij}=w\left(\left[\begin{array}{cc}0& a_{ij}\\ a_{ji}& 0\end{array}\right]\otimes B\right).$ This refines Holbrook's classical bound $w(A\otimes B)\le w(A)\Vert B\Vert$ [J. Reine Angew. Math. 1969], when all entries of $A$ are non-negative. If moreover $a_{ii}\ne 0$ $\forall i$, we prove that $w(A\otimes B)=w(A)\Vert B\Vert$ if and only if $w(B)=\Vert B\Vert .$ 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 $m\ge 1$ that $w(A^{\circ m})\le w^m(A)$ - 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.

Updated the Introduction, added new material from Theorem 4.8 to Corollary 4.14, and also the final Section 7. Now at 22 pages

References in corpus (2)

• Numerical radius inequalities for tensor product of operators
• The $p$-norm of circulant matrices