On the strict monotonicity of spectral radii for classes of bounded positive linear operators

Strict monotonicity of the spectral radii of bounded, positive, ordered linear operators is investigated. It is well-known that under reasonable assumptions, the spectral radii of two ordered positive operators enjoy a non-strict inequality. It is also well-known that a “strict” inequality between operators does not imply strict monotonicity of the spectral radii in general—some additional structure is required. We present a number of sufficient conditions on both the cone and the operators for such a strict ordering to hold which generalise known results in the literature, and have utility in comparison arguments, ubiquitous in positive systems theory.