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Disjointness preserving and diffuse operators. (English) Zbl 0757.47023

The authors consider disjointness preserving operators between Banach lattices and study properties of the collection of all such operators. Given a Riesz space \(E\) and a Dedekind complete vector lattice \(F\), then denote by \({\mathcal L}_ b(E,F)\) the Dedekind complete vector lattice of all order bounded (= regular) operators from \(E\) into \(F\) and by \({\mathcal L}_ n(E,F)\) the band in \({\mathcal L}_ b(E,F)\) of all order bounded, order continuous operators from \(E\) into \(F\). \(\operatorname{Hom}(E,F)\) denotes the set of all Riesz homomorphisms from \(E\) into \(F\), whereas the band generated by \(\operatorname{Hom}(E,F)\) in \({\mathcal L}_ b(E,F)\) is denoted by \({\mathcal H}(E,F)\) and its disjoint complement in \({\mathcal L}_ b(E,F)\) by \({\mathcal D}(E,F)\). The projection of \({\mathcal L}_ b(E,F)\) onto \({\mathcal H}(E,F)\) is denoted by \({\mathcal P}_{EF}\). Then they prove the following results:
Theorem 2.2. Let \(E\) be a vector lattice and \(F\) a Dedekind complete vector lattice. If \(T\in {\mathcal L}_ b(E,F)\), then \(T\in{\mathcal D}(E,F)\) if and only if \[ \inf\bigl\{\bigvee_{i=1}^ n | T| u_ i:\;| f|\leq\bigvee_{j=1}^ m u_ j;\;u_ 1,u_ 2,\dots,u_ n\in E^ +,\;n\in N\bigr\}=0 \] for all \(f\in E\).
Theorem 3.4. If \(S\in {\mathcal L}_ b(E,F)\) and \(T\in{\mathcal L}_ n(F,G)\), then
(i) \({\mathcal P}_{EG}(TS)={\mathcal P}_{EG}(T{\mathcal P}_{EF}(S))\),
(ii) \({\mathcal P}_{EG}({\mathcal P}_{EG}(T)S)={\mathcal P}_{EG}(T){\mathcal P}_{EG}(S)\).
As a consequence of Theorem 3.4 they observe that \({\mathcal P}_ E(TS)={\mathcal P}_ E(T{\mathcal P}_ E(S))\) and \({\mathcal P}_ E({\mathcal P}_ E(T)S)={\mathcal P}_ E(T){\mathcal P}_ E(S)\) for all \(S\in{\mathcal L}_ b(E)\), \(T\in{\mathcal L}_ n(E)\); the latter equality means that the band projection in \({\mathcal L}_ n(E)\) onto \({\mathcal H}_ n(E)\) satisfies the left averaging identity. Then it is proved that the phenomenon of averaging band projection also occurs in some slightly more general situation. Section 4 contains examples and counterexamples illustrating the previous results. For instance, it is shown
1) that any left translation invariant operator \(S\in{\mathcal L}_ b(\ell_ \infty,E)\) belongs to \({\mathcal D}(\ell_ \infty,E)\);
2) there exist vector lattices with only trivial diffuse operators;
3) if \(E\), \(F\) are Banach lattices and \(E^*\) does not contain atoms, there exist non trivial compact lattice homomorphisms.
In section 5 the authors study the band projection in \({\mathcal L}_ b(E)\) onto the principal band generated by a \(T\) belonging to \({\mathcal L}_ b(E)\), obtaining the following result:
Theorem 5.5. Let \(E\) be a Dedekind complete Banach lattice with lower semicontinuous norm and \(T\in\operatorname{Hom}(E)\). Then the band projection \({\mathcal P}_ T\) in \({\mathcal L}_ b(E)\) onto the principal band generated by \(T\) is contractive with respect to the operator norm.
The last section contains some results on the spectral theory of disjointness preserving operators, based on Theorems 3.4 and 5.5.

MSC:

47B60 Linear operators on ordered spaces
47L05 Linear spaces of operators
46A40 Ordered topological linear spaces, vector lattices
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References:

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