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Spectral and asymptotic properties of contractive semigroups on non-Hilbert spaces. (English) Zbl 1399.47117
The author studies the asymptotic behavior of $$C_{0}$$-semigroups $$e^{{\kern 1pt} tA}$$, $$t\geq 0$$, on real Banach spaces. To describe the main result, two definitions are needed. First, a real Banach space $$X$$ is called “extremely non-Hilbert” if it does not isometrically contain a two-dimensional Hilbert space. Second, the semigroup $$e^{{\kern 1pt} tA}$$ is called “weakly asymptotically contractive” if $$\mathop{\lim}\limits_{t\to \infty} \sup | \left. \left\langle e^{{\kern 1pt} tA} x,x'\right. \right\rangle |\; \leq 1$$ for every $$x\in X$$ and $$x'\in X'$$ with $$\| x\|\; =\; \| x'\|\; =1$$. The following theorem is proved: Let $$X$$ be an extremely non-Hilbert real Banach space and let $$e^{{\kern 1pt} tA}$$, $$t\geq 0$$, be a weakly asymptotically contractive $$C_{0}$$-semigroup on $$X$$. Then $$\sigma_{p} (A) \bigcap i{\mathbb R}\subseteq \{ 0\}$$. Here, $$\sigma_{p} (A)$$ is the point spectrum of the generator $$A$$. The author provides a number of interesting corollaries and examples. A single-operator version of the theorem is also proved.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 47A10 Spectrum, resolvent
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