On norm continuity, differentiability and compactness of perturbed semigroups
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On norm continuity, differentiability and compactness of perturbed semigroups A. Boulouz1 · H. Bounit1 · A. Driouich1 · S. Hadd1 Received: 10 May 2019 / Accepted: 26 February 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The main purpose of this paper is to treat semigroup properties like norm continuity, compactness and differentiability for perturbed semigroups in Banach spaces. In particular, we investigate three large classes of perturbations: Miyadera–Voigt, Desch–Schappacher and Staffans–Weiss perturbations. Our approach is mainly based on feedback theory of Salamon–Weiss systems. Our results are applied to abstract boundary integro-differential equations in Banach spaces. Keywords Operator semigroup · Unbounded perturbation · Norm continuity · Compactness · Differentiability · Bergman space · Feedback theory · Integrodifferential equations
1 Introduction In this paper we investigate classical properties like norm continuity, compactness and differentiability for some classes of perturbed semigroups. To be more precise, let X, Z be Banach spaces and (A, D(A)) a generator of a strongly continuous semigroup 𝕋 ∶= (T(t))t≥0 on X such that D(A) ⊂ Z ⊂ X . We introduce a linear operator ̃ X) ̃ where Z̃ and X̃ are Banach spaces carefully chosen in such a way that 𝕃 ∈ L(Z, Communicated by Abdelaziz Rhandi. * S. Hadd [email protected] A. Boulouz [email protected] H. Bounit [email protected] A. Driouich [email protected] 1
Department of Mathematics, Faculty of Sciences, Ibn Zohr University, Hay Dakhla, BP 8106, 80000 Agadir, Morocco
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A + 𝕃 with appropriate domain is well defined and generates a strongly continuous semigroup 𝕋 cl ∶= (T cl (t))≥0 on X (this notation will be justified in Sect. 2). Now the problem to be treated is: do the two semigroups generated by A and A + 𝕃 share the aforementioned properties? As a matter of fact this problem has already been considered by many authors who have some partial answers (depending on the type of perturbations). The class of bounded perturbations, i.e., the case when X = Z̃ = X̃ (so that 𝕃 ∈ L(X) ), is mainly treated by Phillips [23]. He proved that if (T(t))t≥0 is norm continuous (resp. compact) for t > 0 , then the operator (A + 𝕃, D(A)) generates a strongly continuous semigroup (T cl (t))t≥0 on X which is norm continuous (resp. compact) for t > 0 , as well. On the other hand, Phillips constructed a semigroup (T(t))t≥0 which is norm continuous for t > t0 with t0 > 0 (i.e., eventually norm continuous) but the semigroup (T cl (t))t≥0 is not norm continuous for t > t0 . Thus, eventual norm continuity and eventual compactness are, in general, not preserved even under bounded perturbations. It is shown in [7, Proposition III.1.14] that in the case of compact perturbation operator 𝕃 ∈ L(X) , the eventual norm continuity is preserved for the perturbed semigroup whenever the initial semigroup is eventually norm continuous. In 1983, Pazy [21] (see also [24]) showed that the
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