Some spectral properties of operator-valued positive-real functions

摘要

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. (C) 2020 Elsevier B.V. All rights reserved.