### Optimal Domain and Integral Extension of Operators Acting in Fréchet Function Spaces

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ISBN 978-3-8325-4557-4

137 pages, year of publication: 2017

price: 35.00 €

It is known that a continuous linear operator T defined on a Banach function space X(μ) (over a finite measure space (Ω,Σ,μ)) and with values in a Banach space X can be extended to a sort of optimal domain.
Indeed, under certain assumptions on the space X(μ) and the operator T this optimal domain coincides with L

^{1}(m

_{T}), the space of all functions integrable with respect to the vector measure m

_{T} associated with T, and the optimal extension of T turns out to be the integration operator I

_{mT}.
In this book the idea is taken up and the corresponding theory is translated to a larger class of function spaces, namely to Fr\'echet function spaces X(μ) (this time over a σ-finite measure space (Ω,Σ,μ).
It is shown that under similar assumptions on X(μ) and T as in the case of Banach function spaces the so-called ``optimal extension process'' also works for this altered situation.
In a further step the newly gained results are applied to four well-known operators defined on the Fréchet function spaces L

^{p-}([0,1]) resp. L

^{p-}(G) (where G is a compact Abelian group) and L

^{p}_{loc} .