Some Inequalities on the Decomposable Numerical Radii of Matrices

Abstract

Let m and n be positive integers such that $1 \leq m \leq n$. Denote by $\mathbb{C}_{n \times m} $ the set of all $n \times m$ complex matrices. For a matrix $A \in \mathbb{C}_{n \times n} $, its mth decomposable numerical radius is defined and denoted by \[ r_m^ \wedge ( A ) = \max\left\{ |\det ( X^* AX ) | : X \in \mathbb{C}_{n \times m}, X^ * X = I_m \right\}. \] If $m = 1$, it reduces to the classical numerical radius of A, which is denoted by $r( A )$; and $r_n^ \wedge ( A ) = | det ( A ) |$. In this note we prove the inequalities \[ r (A) \equiv r_1^ \wedge (A) \geq r_2^ \wedge ( A )^{1/ 2} \geq \cdots \geq r_{n - 1}^ \wedge ( A )^{1/( n - 1)} \geq r_n^ \wedge ( A )^{1/n} \equiv | \det ( A ) |^{1/n} , \] and \[ \begin{pmatrix} n \\ m \end{pmatrix} r_m^ \wedge ( A ) \geq E_m ( \sigma_1 ( A ), \cdots ,\sigma _n ( A ) ) \] where $E_m ( \cdot )$ denotes the mth elementary symmetric function, and $\sigma _1 ( A ) \geq \cdots \geq \sigma _n ( A )$ are the singular values of A. Complete characterizations of the matrices for which any one of the equalities holds are given.