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Some characterizations of Riesz spaces in the sense of strongly order bounded operators. (English) Zbl 1435.47046

Summary: We investigate some properties of strongly order bounded operators. For example, we prove that if a Riesz space \(E\) is an ideal in \(E^{\sim \sim}\) and \(F\) is a Dedekind complete Riesz space then for each ideal \(A\) of \(E\), \(T\) is strongly order bounded on \(A\) if and only if \(T_A\) is strongly order bounded. We show that the class of strongly order bounded operators satisfies the domination problem. On the other hand, we present two ways for decomposition of strongly order bounded operators, and we give some of their properties. Also, it is shown that \(E\) has order continuous norm or \(F\) has the \(b\)-property whenever each pre-regular operator form \(E\) into \(F\) is order bounded.

MSC:

47B65 Positive linear operators and order-bounded operators
46B40 Ordered normed spaces
46B42 Banach lattices
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References:

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