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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).

MSC:
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
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References:
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