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A new proof and generalizations of Gearhart’s theorem. (English) Zbl 1119.47043
Let \(H\) be a Hilbert space with inner product \((\cdot,\cdot)_H\). Denote by \(AP_b(\mathbb R,H)\) the space of Bohr’s almost periodic functions defined on \(\mathbb R\) with values in \(H\). It is known that, if \(f, g \in AP_b(\mathbb R, H)\), then the limit \[ \langle f,g\rangle := \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T (f(t),g(t))_H\,dt \] exists and defines an inner product. Denote by \(AP(\mathbb R,H)\) the completion of \(AP_b(\mathbb R, H)\) with respect to the norm generated by the inner product. If \(f \in AP(\mathbb R,H)\) and \(\lambda \in \mathbb R\), define \[ a(\lambda,f):= \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T e^{i\lambda t} f(t)\,dt. \] The set \(\{\lambda \in \mathbb R:a(\lambda,f)\neq 0\}\) (at most countable) is called the Bohr spectrum of \(f\).
Let \(A\) be a closed densely defined linear operator in \(H\) and let \(\Lambda\) be a closed subset of \(\mathbb R\) and \(\rho(A)\) be the resolvent set of \(A\). Consider the equation \[ u'(t) = Au(t) + f(t), \;t \in {\mathbb R}.\tag{1} \] Then it is proved that the two following conditions are equivalent: (i) for every \(f\in AP(\mathbb R,H)\) such that \(\sigma(f)\subseteq\Lambda\), there exists a unique mild solution \(u\) in \(AP(\mathbb R,H)\) of (1) such that \(\sigma(u)\subseteq\Lambda\); (ii) \(i\Lambda\subseteq\rho(A)\) and \(\sup_{\lambda\in\Lambda}\|(i\lambda - A)^{-1}\|<\infty\).

47D06 One-parameter semigroups and linear evolution equations
34G10 Linear differential equations in abstract spaces
47N20 Applications of operator theory to differential and integral equations
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI
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