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Isometric factorization of vector measures and applications to spaces of integrable functions | | Olav Nygaard
; José Rodríguez
; | | Date: |
31 Jul 2020 | | Abstract: | Let $X$ be a Banach space, $Sigma$ be a $sigma$-algebra, and $m:Sigma o
X$ be a (countably additive) vector measure. It is a well known consequence of
the Davis-Figiel-Johnson-Pelczýnski factorization procedure that there
exist a reflexive Banach space $Y$, a vector measure $ ilde{m}:Sigma o Y$
and an injective operator $J:Y o X$ such that $m$ factors as $m=Jcirc
ilde{m}$. We elaborate some theory of factoring vector measures and their
integration operators with the help of the isometric version of the
Davis-Figiel-Johnson-Pelczýnski factorization procedure. Along this way, we
sharpen a result of Okada and Ricker that if the integration operator on
$L_1(m)$ is weakly compact, then $L_1(m)$ is equal, up to equivalence of norms,
to some $L_1( ilde m)$ where $Y$ is reflexive; here we prove that the above
equality can be taken to be isometric. | | Source: | arXiv, 2008.00090 | | Services: | Forum | Review | PDF | Favorites | |
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