## Spectral properties and stability of one-parameter semigroups.(English)Zbl 0801.47026

Summary: Let $$A$$ be the infinitesimal generator of a strongly continuous semigroup $$e^{tA}$$ of bounded linear operators in a Banach space $$X$$ with norm $$\| \cdot \|$$, and assume that $$Y \subset X$$ is also a Banach space with norm $$\| \cdot \|_ Y$$ which is stronger than the norm $$\| \cdot \|$$ and $$Y$$ is dense in $$X$$. Moreover, suppose that $$e^{tA}Y \subset Y$$ for $$t \geq 0$$ and $$e^{tA}$$ is also a strongly continuous semigroup in $$Y$$ with the infinitesimal generator $$B$$. We show, when $$e^{tA}$$ is an isometric group, that
(a) if $$\lambda \in \sigma(A)$$, the spectrum of $$A$$, is isolated, then $$\lambda \in \sigma_ p (A)$$, the point spectrum of $$A$$;
(b) if $$\sigma (B) \cap (iR)$$ is countable, then $$\sigma (A) = \sigma (B)$$ and $$\sigma_ p (B)$$ $$(\subset \sigma_ p (A))$$ is nonempty.
As an application of (a) and (b), we show that if $$e^{tA}$$ is uniformly bounded, $$\sigma (B) \cap (iR)$$ is contained in $$\sigma_ c(B)$$ and is countable, then $$\lim_{t \to \infty} e^{tA}x = 0$$ for all $$x \in X$$, where $$\sigma_ c(B)$$ denotes the continuous spectrum of $$B$$.

### MSC:

 47D06 One-parameter semigroups and linear evolution equations
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