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.

中文翻译：

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算子值正实函数的一些谱性质

摘要 我们考虑算子值正实函数 H 并表明 H ( s ) 的点谱、连续谱和残差谱与虚轴的交点不依赖于 s 。特别地，如果 H 是正实数，并且 H ( z ) 对于开右半平面中的某些 z 是可逆的，那么 H ( s ) 对于开右半平面中的所有 s 都是可逆的。此外，我们证明了对应于虚特征值的 H ( s ) 的特征空间不依赖于 s 。还表明 H(s) 的数值范围与虚轴的交点与 s 无关。最后，我们证明，在适当的假设下，将“足够正实”静态输出反馈应用于正实传递函数会导致严格正实闭环系统。