Sufficient conditions for stability in critical cases (Q1593954)

Let \(\rho(A)\) denote the spectral radius of the operator \(A\), \(\sigma(A)\) the spectrum of \(A\), and \(\tau(A)= \max\{\text{Re }\lambda; \lambda\in\Sigma(A)\}\). The author treats \(D\)-system \(x(t+ 1)= Mx(t)\), \(t= 0,1,2,\dots\), and \(C\)-system \((d/dt) x(t)= Kx(t)\), \(0\leq t< \infty\), where \(M\), \(K\) are a linear bounded operators that act in the Hilbert spaces \(H_M\), \(H_K\), respectively. Assume that there exists a unitary operator \(U: K_U\to K_U\), a skew-Hermitean operator \(V: K_V\to K_V\), and linear bounded operators \(Y_U: K_U\to H_M\), \(Y_V: K_V\to H_K\) having bounded inverses such that \(M^* Y_U= Y_U U^*\), \(K^* Y_V= Y_V V^*\). \(\ker Y^*_U\), \(\ker Y^*_V\) are invariant under \(M\), \(K\). He gives the following conditions for stability of the two systems in \(\ker Y^*_U\), \(\ker Y^*_V\): (I) \(\rho(M|\ker Y^*_U)< 1\) is valid if and only if one can find an operator \(W: K_U\to H_M\) such that \(\rho(M- WY^*_U)< 1\) or \(\|M- MY^*_U\|< 1\). (II) \(\tau(K|\ker Y^*_V)< 1\) is valid if and only if one can find an operator \(W: K_V\to H_K\) such that \(\tau(K- WY^*_V)< 1\) or \[ \lim_{0<\varepsilon\to 0}(\|\text{I}+ \varepsilon(K- WY^*_V)\|- \|\text{I}\|)/\varepsilon< 0. \] Here two norms \(\|\cdot\|\) are equivalent to the original ones.