Optimal Domains and Integral Extensions

Part of the Operator Theory: Advances and Applications book series (OT, volume 180)


In the previous chapter we presented a selection of relevant aspects from the theory of vector measures and integration. Following a time honoured practice, we began with a vector measure ? and ended up with operators defined on the spaces Lp(v) and Lp/w(v). In this chapter we reverse this line of development in a certain sense. Namely, we begin with an operator T : X(μ) → E, defined on some σ-order continuous q-B.f.s. X(μ) and taking values in a Banach space E, and produce from it the E-valued vector measure mT : A → T(XA). This, in turn, has associated with it the B.f.s. L1(mT) which is σ-o.c. and has the desirable property that every function from X(μ) belongs to L1(MT) and
$$ T(f) = \int_\Omega {f dm_T } , f \in X(\mu ) \subseteq L^1 (m_T ). $$


Banach Space Banach Lattice Vector Measure Fredholm Operator Continuous Linear Operator 
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© Birkhäuser Verlag AG 2008

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