On positive strictly singular operators and domination (Q1412292)

Let \(X, Y\) be Banach spaces. An operator \(T: X \rightarrow Y\) is called strictly singular if and only if every infinite-dimensional subspace \(M\) of \(X\) contains an infinite-dimensional subspace \(N\) such that the restriction of \(T\) to \(N\) is compact. The authors study the problem of domination by strictly singular operators. In some very special cases, the problem has an affirmative answer. They provide examples showing that, in general, the problem of domination for strictly singular operators has a negative answer. The main contribution is the following result. Theorem: Let \(E\) and \(F\) be order continuous Banach lattices such that \(E\) has the subsequence splitting property and the dual norm on \(E'\) is order continuous. If \(T\) is a positive strictly singular operator from \(E\) to \(F\), then every operator in \([0,T]\) is strictly singular. The authors provide an example to show that the order continuity of \(F\) is indispensible.