Optimal Domain and Integral Extension of Operators Acting in Fréchet Function Spaces
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 L1
), the space of all functions integrable with respect to the vector measure mT
associated with T, and the optimal extension of T turns out to be the integration operator ImT
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 Lp-
([0,1]) resp. Lp-
(G) (where G is a compact Abelian group) and Lploc