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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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