# Spectral criteria for solutions of evolution equations and comments on reduced spectra - Mathematics > Functional Analysis

Abstract: We revisit the notion of reduced spectra $sp {\Cal {F}} \phi$ for boundedmeasurable functions $\phi \in L^{\infty} J,\Bbb{X}$, ${\Cal {F}}\subsetL^1 {loc}J,\Bbb{X}$. We show that it can not be obtained via Carleman spectraunless $\phi\in BUCJ,\Bbb{X}$ and ${\Cal {F}} \subset BUCJ,\Bbb{X}$. Insection 3, we give two examples which seem to be of independent interest forspectral theory. In section 4, we prove a spectral criteria for bounded mildsolutions of evolution equation * $\frac{d ut}{dt}= A ut + \phi t$,$u0=x\in \Bbb{X}$, $t\in {J}$, where$A$ is a closed linear operator on $\Bbb{X}$ and $\phi\in L^{\infty} {J},\Bbb{X}$ where ${J} \in\{ +, \}$.

Author: Bolis Basit, Hans Günzler

Source: https://arxiv.org/