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

Original · EN

We revisit the notion of reduced spectra sp F (ϕ) for bounded measurable functions ϕ∈ L∞ (J,X), F⊂ L¹loc(J,X). We show that it can not be obtained via Carleman spectra unless ϕ∈ BUC(J,X) and F ⊂ BUC(J,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 (*) d u(t)/dt= A u(t) + ϕ(t), u(0)=x∈ X, t∈ J, where A is a closed linear operator on X and ϕ∈ L∞ (J, X) where J ∈{+,}.