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

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

2010-06-10

arxiv.org/abs/1006.2169v2

Mathematics

Functional Analysis

16 pages

Scientific paper

We revisit the notion of reduced spectra $sp_{\Cal {F}} (\phi)$ for bounded measurable functions $\phi \in L^{\infty} (J,\Bbb{X})$, ${\Cal {F}}\subset L^1_{loc}(J,\Bbb{X})$. We show that it can not be obtained via Carleman spectra unless $\phi\in BUC(J,\Bbb{X})$ 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{d u(t)}{dt}= A u(t) + \phi (t) $, $u(0)=x\in \Bbb{X}$, $t\in {J}$, where $A$ is a closed linear operator on $\Bbb{X}$ and $\phi\in L^{\infty} ({J}, \Bbb{X})$ where $ {J} \in\{\r_+,\r\}$.

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