On Monotone Linear Operators and the Spectral Radius of Their Representing Matrices
H.H. Tigelaar · SIAM Journal on Matrix Analysis and Applications · 1991
In this paper, linear operators on the space of $p \times p$ matrices are considered. Such linear operators can be represented by $p^2 \times p^2 $ matrices. In particular, sums of Kronecker products occur as representing matrices. Let the linear operators $\mathcal{L}_S $ and $\mathcal{L}_U $ be represented by the matrices S and U, where U is of the form $U = \sum {A_k } \otimes \bar A_k $. It is shown that, in order that $\mathcal{L}_U ( X )\leqq \mathcal{L}_S ( X )$ for all positive-semidefinite X, it is necessary that the spectral radii of U and S satisfy the inequality $\rho ( U )\leqq \rho ( S )$.