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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 \(\sup_{\lambda\in\Lambda}\|(i\lambda - A)^{-1}\|<\infty\).

MSC:
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
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