Vol. 66, No. 2, 1976

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Some remarks on convolution equations for vector-valued distributions

Hector O. Fattorini

Vol. 66 (1976), No. 2, 347–371

Let E, X be two complex Banach spaces, (E;X) the space of all linear bounded operators from E into X endowed with its usual norm. We denote by 𝒟′(E) the space of distributions with values in E defined in −∞ < t < and by 𝒟0(E) the subspace of 𝒟′(E) consisting of all T ∈𝒟′(E) with support in t 0.

Given P ∈𝒟0((X;E)) we examine the following problems.

(I). Does P have a convolution inverse with support in t 0, that is, is there S ∈𝒟0((E;X)) satisfying

P ∗S = δ⊗ I,S ∗P = δ⊗ J

where I (resp. J) denotes the identity operator in E (resp. X) and δ is the Dirac delta?

(II). In case the answer to (I) is affirmative, what properties of S can be deduced from properties of P and vice versa?

Mathematical Subject Classification 2000
Primary: 46F10
Secondary: 47D05
Received: 8 May 1975
Revised: 3 May 1976
Published: 1 October 1976
Hector O. Fattorini