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

2010/06/10 by Bolis Basit, Basit, Bolis, Hans Günzler +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA

paper · pdf · doi:10.48550/arxiv.1006.2169

16 pages

arxiv created 2010/07/08 · arxiv updated 2010/07/09

Abstract

We revisit the notion of reduced spectra sp_\Cal F (ϕ) for bounded measurable functions ϕ∈ L^∞ (J,\BbbX), \Cal F⊂ L¹_loc(J,\BbbX). We show that it can not be obtained via Carleman spectra unless ϕ∈ BUC(J,\BbbX) and \Cal F ⊂ BUC(J,\BbbX). 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 (*) (d u(t))/(dt)= A u(t) + ϕ(t) , u(0)=x∈ \BbbX, t∈ J, where A is a closed linear operator on \BbbX and ϕ∈ L^∞ (J, \BbbX) where J ∈\\r₊,\r\.

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