Order-continuous component of a majorized operator
A real vector space X equipped with a map \(| \cdot |:X\to E\), where E is a K-space \((=\) order complete vector lattice), such that (1) \(| x| \geq 0\); \(| x| =0\) if and only if \(x=0\) (x\(\in X)\); (2) \(| x+y| \leq | x| +| y|\) (x,y\(\in X)\); (3) \(| \lambda x| =| \lambda | | x|\) (x\(\in X,\lambda \in R)\); is called a K-normed space. The norm \(| \cdot |\) is called decomposable if, for every \(x\in X\) and \(e_ 1,e_ 2\in E\) with \(e_ 1\wedge e_ 2=0\) and \(| x| =e_ 1+e_ 2\), there exist \(x_ 1,x_ 2\in X\) such that \(x=x_ 1+x_ 2\) and \(| x_ i| =e_ i\), \(i=1,2\). The K-normed space X is called bo-complete if, for every net \((x_{\alpha})\) in X with \(o-\lim | x_{\alpha}- x_{\beta}| =0\), there exists \(x\in X\) with \(o-\lim | x- x_{\alpha}| =0\). Let Y be another K-normed space, the norm of which takes values in a K-space F. A linear operator T:X\(\to Y\) is called bo- continuous if, for every net \((x_{\alpha})\) in X with \(o-\lim | x_{\alpha}| =0\), we have \(o-\lim | Tx_{\alpha}| =0\). We call T majorized if there exists a positive operator S:E\(\to F\) such that \(| Tx| \leq S(| x|)\) for all \(x\in X\). The vector space of such operators is denoted by M(X,Y). (All these notions go back to L. V. Kantorovich.) Generalizing the Yosida-Hewitt decomposition theorem, the first author and \textit{V. Z. Strizhevskij} [Trudy Inst. Mat. 7, 132-158 (1987)] proved that if X and Y are bo-complete, then every \(T\in M(X,Y)\) can be uniquely represented as follows: \(T=T_ n+T_ s\), where \(T_ n\), \(T_ s\in M(X,Y)\) and \(T_ n\) is bo-continuous and \(T_ s\) is singular in an appropriate sense. In the present paper this result is generalized in two directions by relaxing the assumptions on X and Y and introducing a more general notion of continuity. Furthermore, two formulas for the bo-continuous part of T, both involving iterated limits, are exhibited. Those formulas have origin in measure theory, and the authors' approach uses some ideas of T. Traynor. The details are highly technical. \{Reviewer's remark: For related Yosida-Hewitt type decompositions in the order setting see \textit{K. D. Schmidt}, Jordan decompositions of generalized vector measures, preprint 1988.\}
scientific article; zbMATH DE number 4048455
scientific article