Some spectral properties of operator-valued positive-real functions

Abstract

We consider operator-valued positive-real functions H and show that the intersections of the point, continuous and residual spectra of H ( s ) with the imaginary axis do not depend on s . In particular, if H is positive real and H ( z ) is invertible for some z in the open right-half plane, then H ( s ) is invertible for all s in the open right-half plane. Furthermore, we prove that the eigenspace of H ( s ) corresponding to an imaginary eigenvalue does not depend on s . It is also shown that the intersection of the numerical range of H ( s ) with the imaginary axis is independent of s . Finally, we prove that, under suitable assumptions, application of a “sufficiently positive-real” static output feedback to a positive-real transfer function leads to a strictly positive-real closed-loop system.