an:06658790
Zbl 1399.47117
Gl??ck, Jochen
Spectral and asymptotic properties of contractive semigroups on non-Hilbert spaces
EN
J. Oper. Theory 76, No. 1, 3-31 (2016).
00361065
2016
j
47D06 47A10
asymtotics of contractive semigroups; peripheral spectrum
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.
Khristo N. Boyadzhiev (Ada)