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On the ring ideal generated by a positive operator. (English) Zbl 0588.47044
Let \(T: E\to F\) be an operator between two Banach lattices. The ring ideal Ring(T) generated by T is the norm closure in L(E,F) \((=the\) Banach space of all norm bounded operators from E into F) of the vector subspace consisting of all operators of the form \(\sum^{n}_{i=1}R_ iTS_ i\), where \(S_ i\in L(E)\) and \(R_ i\in L(F)\), \(i=1,...,n\). If F is Dedekind complete and T is an order bounded (i.e., regular) operator, then the order ideal \({\mathcal A}_ T\) generated by T in \({\mathcal L}_ b(E,F)\) \((=the\) Riesz space of all order bounded operators from E into F) consists of all operators \(S\in {\mathcal L}_ b(E,F)\) for which there exists some \(\lambda >0\) with \(| S| \leq \lambda | T|\). In this case we have \({\mathcal A}_ T\subseteq L(E,F).\)
Assume now that T is a positive operator. This paper studies various properties between the ring and order ideals generated by T. The major results.
1. Let E be either \(\sigma\)-Dedekind complete or with a quasi-interior point and let F be Dedekind complete. If T has order continuous norm (i.e., if \(T\geq T_ n\downarrow 0\) implies \(\| T_ n\| \downarrow 0)\), then \({\mathcal A}_ T\subseteq Ring(T)\) holds.
2. Let \(E=F\) and let another positive operator \(S: E\to E\) satisfy \(0\leq S\leq T\). Then we have:
a) If S and its adjoint are both semicompact (an operator \(R: E\to F\) is semicompact whenever for each \(\epsilon >0\) there exists some \(u\in F^+\) such that \(\| (| Rx| -u)^+\| <\epsilon\) for all \(\| x\| \leq 1)\), then \(S^ 3\in Ring(T).\)
b) If E has order continuous norm and S is semicompact, then \(S^ 2\) belongs to Ring(T).
3. Let \(E=F\) and let two other positive operators S,R: \(E\to E\) satisfy \(0\leq S\leq T\leq R\). If T is a compact operator, then:
a) \(S^ 3\) belongs to Ring(T) (and hence \(S^ 3\) is compact); and
b) \(T^ 3\) belongs to Ring(R).
A simpler presentation and more results can be found in Section 18 of the authors’ recent book ”Positive Operators”, Pure and Applied Math. Series 119, Academic Press (1986).

47B60 Linear operators on ordered spaces
46A40 Ordered topological linear spaces, vector lattices
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
46B42 Banach lattices
Full Text: DOI
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