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Nuclear operators on Banach function spaces
Positivity ( IF 0.8 ) Pub Date : 2020-10-06 , DOI: 10.1007/s11117-020-00787-1
Marian Nowak

Let X be a Banach space and E be a perfect Banach function space over a finite measure space \((\Omega ,\Sigma ,\lambda )\) such that \(L^\infty \subset E\subset L^1\). Let \(E'\) denote the Köthe dual of E and \(\tau (E,E')\) stand for the natural Mackey topology on E. It is shown that every nuclear operator \(T:E\rightarrow X\) between the locally convex space \((E,\tau (E,E'))\) and a Banach space X is Bochner representable. In particular, we obtain that a linear operator \(T:L^\infty \rightarrow X\) between the locally convex space \((L^\infty ,\tau (L^\infty ,L^1))\) and a Banach space X is nuclear if and only if its representing measure \(m_T:\Sigma \rightarrow X\) has the Radon-Nikodym property and \(|m_T|(\Omega )=\Vert T\Vert _{nuc}\) (= the nuclear norm of T). As an application, it is shown that some natural kernel operators on \(L^\infty \) are nuclear. Moreover, it is shown that every nuclear operator \(T:L^\infty \rightarrow X\) admits a factorization through some Orlicz space \(L^\varphi \), that is, \(T=S\circ i_\infty \), where \(S:L^\varphi \rightarrow X\) is a Bochner representable and compact operator and \(i_\infty :L^\infty \rightarrow L^\varphi \) is the inclusion map.



中文翻译:

Banach功能空间上的核算子

X为Banach空间,E为有限度量空间\((\ Omega,\ Sigma,\ lambda)\)上的完美Banach函数空间,使得\(L ^ \ infty \ subset E \ subset L ^ 1 \ )。让\(E '\)表示的Köthe双é\(\ tau蛋白(E,E')\)代表的自然麦基拓扑ê。结果表明,在局部凸空间\((E,\ tau(E,E'))\)和Banach空间X之间的每个核算子\(T:E \ rightarrow X \)都是Bochner可表示的。特别地,我们获得线性运算符\(T:L ^ \ infty \ rightarrow X \)当且仅当它的表示度量\(m_T:\ Sigma \ rightarrow )时,局部凸空间\((L ^ \ infty,\ tau(L ^ \ infty,L ^ 1))\)和Banach空间X之间是核的X \)具有Radon-Nikodym属性,并且\(| m_T |(\ Omega)= \ Vert T \ Vert _ {nuc} \)(= T的核范数)。作为一个应用,证明了\(L ^ \ infty \)上的某些自然内核运算符是核的。此外,表明每个核算子\(T:L ^ \ infty \ rightarrow X \)都通过某个Orlicz空间\(L ^ \ varphi \)进行因式分解,即\(T = S \ circ i_ \ infty \),其中\(S:L ^ \ varphi \ rightarrow X \)是Bochner可表示且紧凑的运算符,\(i_ \ infty:L ^ \ infty \ rightarrow L ^ \ varphi \)是包含关系图。

更新日期:2020-10-07
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