Let $A=(a_{j,k})_{j,k \ge 1}$ be a non-negative matrix. In this paper, we characterize those $A$ for which $\|A\|_{E, F}$ are determined by their actions on decreasing sequences, where $E$ and $F$ are suitable normed Riesz spaces of sequences. In particular, our results can apply to the following spaces: $\ell_p$, $d(w,p)$, and $\ell_p(w)$. The results established here generalize ones given by Bennett; Chen, Luor, and Ou; Jameson; and Jameson and Lashkaripour.

We use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of $p$-convex, $p$-concave and positive $p$-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators.

Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calder{\'o}n--Lozanovski\u\i\ construction. Factorization theorems for operators in spaces more general than the Lebesgue $L^{p}$ spaces are investigated. It is natural to extend the Gagliardo theorem on the Schur test and the Rubio de~Francia theorem on factorization of the Muckenhoupt $A_{p}$ weights to reflexive Orlicz spaces. However, it turns out that for the scales far from $L^{p}$-spaces this is impossible. For the concrete integral operators it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces are not valid. Representation theorems for the Calder{\'o}n--Lozanovski\u\i\ construction are involved in the proofs.

A complete description of symmetric spaces on a separable measure space with the Dunford-Pettis property is given. It is shown that $\ell^1$, $c_0$ and $\ell^{\infty}$ are the only symmetric sequence spaces with the Dunford-Pettis property, and that in the class of symmetric spaces on $(0, \alpha)$, $0 < \alpha \leq \infty$, the only spaces with the Dunford-Pettis property are $L^1$, $L^{\infty}$, $L^1 \cap L^{\infty}$, $L^1 + L^{\infty}$, $(L^{\infty})^\circ$ and $(L^1 + L^{\infty})^\circ$, where $X^\circ$ denotes the norm closure of $L^1 \cap L^{\infty}$ in $X$. It is also proved that all Banach dual spaces of $L^1 \cap L^{\infty}$ and $L^1 + L^{\infty}$ have the Dunford-Pettis property. New examples of Banach spaces showing that the Dunford-Pettis property is not a three-space property are also presented. As applications we obtain that the spaces $(L^1 + L^{\infty})^\circ$ and $(L^{\infty})^\circ$ have a unique symmetric structure, and we get a characterization of the Dunford-Pettis property of some K\"othe-Bochner spaces.