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The asymptotic behaviour of semigroups of linear operators. (English) Zbl 0905.47001
Operator Theory: Advances and Applications. 88. Basel: Birkhäuser. xii, 236 p. (1996).
Let $${\mathbf T}=(T_t)_{t\geq 0}$$ be a strongly continuous semigroup of bounded linear operators on the Banach space $$X$$. The following three notions of stability are fundamental: $$\mathbf T$$ is called uniformly exponentially stable if $$\| T_t\|\leq M\exp(-\omega t)$$ for some appropriate $$M>0$$ and $$\omega>0$$. It is called exponentially stable if $$\| T_t(x)\|\leq M\exp(-\omega t)\| x\|_{D(A)}$$ for some $$M$$, some $$\omega$$ and for all $$x$$ in the domain of definition $$D(A)$$ of the infinitesimal generator $$A$$ of $$\mathbf T$$. It is called uniformly stable if $$\lim_{t\to\infty}\| T_t(x)\|=0$$ for all $$x$$.
There is a close connection between the spectrum $$\sigma(A)$$ of the generator $$A$$ of $$\mathbf T$$ and these notions of stability. To this end a lot of spectral bounds are defined, which allow to give criteria for stability. Two of these bounds are $$\omega_0({\mathbf T})=\lim_{t\to\infty}{\log(\| T_t\|)\over t}$$, $$\omega_1({\mathbf T})=\inf\{\omega:\| T_t(x)\|\leq M\exp(-\omega t)\| x\|_{D(A)}$$ for all $$x\in D(A)\}$$. Besides the spectral bound $$s(A)=\inf\{\text{Re}(\lambda):\lambda\in\rho(A)\}$$ the number $$s_0(A)=\inf\{\omega:\{\lambda:\text{Re}(\lambda)>\omega\}\subset\rho(A)$$ and $$\sup_{\text{Re}(\lambda)>\omega} \|(\lambda-A)^{-1}\|<\infty\}$$ plays also a fundamental role in stability theory.
The book under review is an excellent monograph on the connection between spectral theory and stability. It contains all of the striking new results in this field including the famous theorem of Arendt-Batty-Lyubich-Phong on the connection between the countability of the peripheral spectrum and the stability of the semigroup. The reader will find also the famous result of L. Weis on the equality of $$s(A)$$ and $$\omega({\mathbf T})$$ for positive semigroups on $$L^p$$-spaces.
The book is divided into 5 chapters: Spectral and growth bounds, spectral mapping theorems, uniform exponential stability, boundedness of the resolvent, countability of the unitary spectrum. An appendix contains an overview over different special topics, e.g. Banach lattices and Banach function spaces.
The list of references is complete, it contains even references to important preprints. The style is clear and elegant. So this book can be highly recommended to all who work on this field of research.

##### MSC:
 47-02 Research exposition (monographs, survey articles) pertaining to operator theory 47D06 One-parameter semigroups and linear evolution equations 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations 93D20 Asymptotic stability in control theory 47A10 Spectrum, resolvent