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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.

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