# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Wolfgang Arendt, Frank Räbiger, and Ahmed Sourour, Spectral properties of the operator equation \?\?+\?\?=\?, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 178, 133 – 149. · Zbl 0826.47013 [2] Larry Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc. 236 (1978), 385 – 394. · Zbl 0326.47038 [3] I. Herbst, The spectrum of Hilbert space semigroups, J. Operator Theory 10 (1983), no. 1, 87 – 94. · Zbl 0535.47024 [4] James S. Howland, On a theorem of Gearhart, Integral Equations Operator Theory 7 (1984), no. 1, 138 – 142. · Zbl 0535.47025 [5] B. M. Levitan and V. V. Zhikov, Almost periodic functions and differential equations, Cambridge University Press, Cambridge-New York, 1982. Translated from the Russian by L. W. Longdon. · Zbl 0499.43005 [6] W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, and U. Schlotterbeck, One-parameter semigroups of positive operators, Lecture Notes in Mathematics, vol. 1184, Springer-Verlag, Berlin, 1986. [7] Jan Prüss, On the spectrum of \?$$_{0}$$-semigroups, Trans. Amer. Math. Soc. 284 (1984), no. 2, 847 – 857. · Zbl 0572.47030 [8] Quoc Phong Vu and E. Schüler, The operator equation \?\?-\?\?=\?, admissibility, and asymptotic behavior of differential equations, J. Differential Equations 145 (1998), no. 2, 394 – 419. · Zbl 0918.34059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.