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 \(B\).


47D06 One-parameter semigroups and linear evolution equations
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