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

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 $c_{ij}=w(\left[\begin{array}{cc}0& a_{ij}\\ a_{ji}& 0\end{array}\right]\otimes B)\mathrm{.}$ This refines Holbrook’s classical bound $w(A\otimes B)\le w(A)\mathrm{\parallel }B\mathrm{\parallel }$ [J. Reine Angew. Math. 1969], when all entries of $A$ are non-negative. If moreover $a_{ii}\mathrm{\ne }0$ $\mathrm{\forall }i$, we prove that $w(A\otimes B)=w(A)\mathrm{\parallel }B\mathrm{\parallel }$ if and only if $w(B)=\mathrm{\parallel }B\mathrm{\parallel }\mathrm{.}$ 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 ${\mathrm{\ell }}_p$ 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.