On some vector lattices of operators and their finite elements (Q1007092)

Various collections of operators between vector lattices \(E\) and \(F\) are studied with the goal of ascertaining whether the collections are vector lattices or, in appropriate cases, Banach lattices. Examples of these types of results include: For \(E\) and \(F\) Banach lattices, the collection of all compact operators endowed with the regular norm, where \(F\) an \(AM\)-space, is a Banach lattice. The collection of all the weakly compact operators, where \(E\) is an \(AL\)-space and \(F\) has property W1 (if \(A\) is weakly compact, then so is \(\{|x| : x\in A\}\)), is a Banach lattice. The relationship between finite rank operators and finite elements is explored. Recall that an element \(\varphi\) in a vector lattice is called finite if there is a \(z\) so that, for any \(x\), there is a constant \(c_x\) so that \(|x|\wedge n|\varphi|\leq c_x z\). In this context, it is established, among numerous other results, that if \(E\) is a Banach lattice and \(F\) is an AM-space with order unit, then the operator \(\varphi \otimes y\), for \(\varphi\in E'\) and \(y\neq 0 \in F\), is finite in the compact operators if and only if \(\varphi\) is finite in \(E'\).