Spectral criteria for solutions of evolution equations and comments on reduced spectra

Abstract

We revisit the notion of reduced spectra $sp_{\Cal {F}} (ϕ)$ for bounded measurable functions $\varphi \in L^{\infty }(J,\mathbb{X})$, ${\Cal {F}}\subset L^1_{loc}(J,\Bbb{X})$. We show that it can not be obtained via Carleman spectra unless $\varphi \in BUC(J,\mathbb{X})$ and ${\Cal {F}} \subset BUC(J,\Bbb{X})$. In section 3, we give two examples which seem to be of independent interest for spectral theory. In section 4, we prove a spectral criteria for bounded mild solutions of evolution equation (*) $\frac{du(t)}{dt}=Au(t)+\varphi (t)$, $u(0)=x\in \mathbb{X}$, $t\in J$, where $A$ is a closed linear operator on $\mathbb{X}$ and $\varphi \in L^{\infty }(J,\mathbb{X})$ where $ {J} \in\{\r_+,\r\}$.

16 pages

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