Optimal Domain and Integral Extension of Operators Acting in Frechet Function Spaces
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 co...
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| Main Author: | |
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| Format: | Electronic Book Chapter |
| Language: | English |
| Published: |
Logos Verlag Berlin
2017
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| Subjects: | |
| Online Access: | DOAB: download the publication DOAB: description of the publication |
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| Summary: | 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(mT), 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 . |
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| ISBN: | /doi.org/10.30819/4557 9783832545574 |
| Access: | Open Access |