Ideal structure and factorization properties of the regular kernel operators (Q830332)

Let \(X,Y\) be two Banach lattices such that \(X_+,Y_+\) represent their respective positive cones. We say that a linear map \(T:X\to Y\) is positive when the image of \(X_+\) (by \(T\)) is included in \(Y_+\) and \(T\) is regular if it is expressed as a linear combination of positive maps. Let \(\mathcal{L}^r(X,Y)\) be the linear space of all \(Y\)-valued regular maps on \(X\). Let \(E,F\) be Banach lattices and \(L\in \mathcal{L}^r(E,F) \), we say that \(T\in \mathcal{L}^r(X,Y)\) is an \(r\)-factors through \(L\) if there are \((R,S)\in \mathcal{L}^r(F,Y)\times \mathcal{L}^r(X,E)\) such that \(T=RLS\). Let \((\Omega,\Sigma,\mu)\) be a \(\sigma\)-finite measure and \(L_p(\mu)\) be the space of \(p\)-integrable functions w.r.t the measure \(\mu\). The authors state sufficient conditions for an \(r\)-factors through \(L_p(\mu)\) to be an \(r\)-factors through \(l_p\).