Estimate of the stability radius of infinite-dimensional systems (Q1594380)

For a compact operator \(A\) on a real Hilbert space \(H\) with spectrum contained in the open left-half plane, the authors study the question when the perturbed operator \(-\tau I+ A+B\) still has its spectrum in the open left half plane. First, they prove that the supremum over the real part of the spectrum of a compact operator \(Q\) equals \(\inf_{T\in{\mathcal L}(H)} \sup_{\|x\|= 1}\langle T^{-1}QTX, x\rangle\). Using this result they obtain a condition on \(B\) which guarantees that the spectrum of \(-\tau I+ A+ B\) has its spectrum in the open left half plane. The condition on \(B\) is in terms of the Hilbert-Schmidt norm, i.e., \(\sqrt{\sum_{k,l}\langle Be_k, e_l\rangle^2}\), where \(\{e_k\}\) is an orthonormal basis of \(H\).