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On the complex inversion formula and admissibility for a class of Volterra systems. (English) Zbl 1291.44001

Summary: This paper studies Volterra integral evolution equations of convolution type from the point of view of complex inversion formula and the admissibility in the Salamon-Weiss sens. We first present results on the validity of the inverse formula of the Laplace transform for the resolvent families associated with scalar Volterra integral equations of convolution type in Banach spaces, which extends and improves the results of E. Hille and R. S. Phillips [Functional analysis and semigroups. Rev. ed. Colloquium Publications. 31. Providence, R. I. American Mathematical Society (AMS). XII, 808 p. (1957; Zbl 0078.10004)] and I. Cioranescu and C. Lizama [Arch. Math. 81, No. 2, 182–192 (2003; Zbl 1043.44002); Lemma 5], respectively, including the stronger version for a class of scalar Volterra integrodifferential equations of convolution type on unconditional martingale differences UMD spaces, provided that the leading operator generates a \(C_0\)-semigroup. Next, a necessary and sufficient condition for \(L^p\)-admissibility \((p\in [1,\infty[)\) of the system’s control operator is given in terms of the UMD-property of its underlying control space for a wider class of Volterra integrodifferential equations when the leading operator is not necessarily a generator, which provides a generalization of a result known to hold for the standard Cauchy problem [H. Bounit et al., Integral Equations Oper. Theory 68, No. 4, 451–472 (2010; Zbl 1204.47047), Proposition 3.2].

MSC:

44A10 Laplace transform
34G10 Linear differential equations in abstract spaces
45D05 Volterra integral equations
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