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Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. (English) Zbl 0593.34048
Let A be the infinitesimal generator of the $$C_ 0$$-semigroup S(t). It is well known that the spectral condition (1) $$\sup \{Re \lambda,\lambda \in \sigma (A)\}<-\delta,$$ $$\delta >0$$, is not sufficient for S(t) to decay exponentially (to be exponentially stable). See e.g. A. Pazy’s book, p. 117. The author shows that even if we assume in addition to (1), that $$(\lambda I-A)^{-1}$$ is compact, S(t) may not decay exponentially. Thus he points out that an existing result in this direction is not true. In the last part of the paper he gives necessary and sufficient conditions, in terms of spectral properties of A, for the exponential stability of S(t). Of course, the condition $$\sup \{Re \lambda <- \delta;\lambda \in \sigma (A)\}$$ appearing in the statement of Theorems A and B has to be formulated in the standard (and better) form (1) above. We may conclude that the author studies some fundamental relationships between the spectral properties of A and the exponential stability of S(t).
Reviewer: N.H.Pavel

##### MSC:
 34D05 Asymptotic properties of solutions to ordinary differential equations 35B40 Asymptotic behavior of solutions to PDEs 37-XX Dynamical systems and ergodic theory 47A50 Equations and inequalities involving linear operators, with vector unknowns 34A30 Linear ordinary differential equations and systems, general 34G10 Linear differential equations in abstract spaces