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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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