Optimal domain and integral extension of operators acting in Fréchet function spaces

It is known that a continuous linear operator T defined on a Banach function space X(μ) (over a finite measure space ( Omega,§igma,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(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₊m₊T. In this book the idea is taken up and the corresponding theory is translated to a larger class of function spaces, namely to Fréchet function spaces X(μ) (this time over a σ-finite measure space ( Omega,§igma,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₊ textloc( mathbbR).