Abstract
Let {P(t): t≥0} be a strongly continuous semigroup on a Banach space X and let |\| be a continuous norm on X such that |P(t)x|≤exp(ωt)|x|, X∈X, t≥0. Let C be a |\|-closed convex subset of X and suppose that for every x in D(A) there exists a sequence (xn : n ε ℕ) in D(A) with the following properties: lim|x−xn|=0, lim|Ax−Axn|=0 and every xn has a best approximation in C (with respect to |\|) which belongs to D(A). Then P(t)C⊂C for all t≥0 if and only if, for every v in C∩D(A), the vector Av belongs to the |\|-closure of [0, ∞) (C-V).
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Van Casteren, J.A. Invariant subsets of strongly continuous semigroups. Integr equ oper theory 7, 884–892 (1984). https://doi.org/10.1007/BF01195871
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DOI: https://doi.org/10.1007/BF01195871