Optimal Domain and Integral Extension of Operators Acting in Frechet Function Spaces - Blaimer, Bettina (Logos Verlag Berlin, 2017)
It is known that a continuous linear operator T defined on a Banach function space X(Î¼) (over a finite measure space ( Omega,§igma,Î¼)) 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(Î¼) and the operator T this optimal domain coincides with LÂ±(mâ T), the space of all functions integrable with respect to the vector measure mâ T associated with T, and the optimal extension of T turns out to be the integration operator Iâ mâ 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(Î¼) (this time over a Ï -finite measure space ( Omega,§igma,Î¼)). It is shown that under similar assumptions on X(Î¼) 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^pâ textloc( mathbbR).