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Let E, X be complex Banach
spaces, (E,X) the space of linear operators from E into X equipped with its usual
norm. We denote by 𝒟′(E) the space of E-valued distributions defined in
−∞ < t < ∞ and by 𝒟′0(E) the subspace thereof consisting of distributions with
support in t ≥ 0. A distribution P ∈𝒟′0((X;E)) is said to have a convolution inverse
(in symbols, P ∈𝒟′0((E;X))−1 or simply P ∈𝒟′0−1) if there exists S ∈𝒟′0((E;X))
such that
 | (1) |
where δ is the Dirac measure and I (resp. J) denotes the identity operator in E
(resp. X). We examine the problem of characterizing those P which possess a
convolution inverse S = P−1 being smooth in various senses: infinitely differentiable,
in a quasi-analytic class, analytic, etc.
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