Some properties of orthogonally additive homogeneous polynomials on Banach lattices

Z. A. Kusraeva ; S. N. Siukaev

Vladikavkazskij matematičeskij žurnal, Tome 22 (2020) no. 4, pp. 92-103

Résumé

Let $E$ and $F$ be Banach lattices and let $\mathcal{P}_o({}^s E,F)$ stand for the space of all norm bounded orthogonally additive $s$-homogeneous polynomial from $E$ to $F$. Denote by $\mathcal{P}_o^r({}^s E,F)$ the part of $\mathcal{P}_o({}^s E,F)$ consisting of the differences of positive polynomials. The main results of the paper read as follows. Theorem 3.4. Let $s\in\mathbb{N}$ and $(E,\|\cdot\|)$ is a $\sigma$-Dedekind complete $s$-convex Banach lattice. The following are equivalent: $(1)$ $\mathcal{P}_o({}^s E,F)\equiv\mathcal{P}_o^r({}^s E,F)$ for every $AM$-space $F$. $(2)$ $\mathcal{P}_o({}^s E,c_0)=\mathcal{P}^r_o({}^s E,F)$ for every $AM$-space $F$. $(3)$ $\mathcal{P}_o({}^s E,c_0)=\mathcal{P}^r_o({}^s E,c_0)$. $(4)$ $\mathcal{P}_o({}^s E,c_0)\equiv\mathcal{P}_o^r({}^s E,c_0)$. $(5)$ $E$ is atomic and order continuous. Theorem 4.3. For a pair of Banach lattices $E$ and $F$ the following are equivalent: $(1)$ $\mathcal{P}_o^r({}^s E,F)$ is a vector lattice and the regular norm $\|\cdot\|_r$ on $\mathcal{P}_o^r({}^s E,F)$ is order continuous. $(2)$ Each positive orthogonally additive $s$-homogeneous polynomial from $E$ to $F$ is $L$- and $M$-weakly compact. Theorem 4.6. Let $E$ and $F$ be Banach lattices. Assume that $F$ has the positive Schur property and $E$ is $s$-convex for some $s\in\mathbb{N}$. Then the following are equivalent: $(1)$ $(\mathcal{P}_o^r({}^s E,F),\|\cdot\|_r)$ is a $K B$-space. $(2)$ The regular norm $\|\cdot\|_r$ on $\mathcal{P}_o^r({}^s E,F)$ is order continuous. $(3)$ $E$ does not contain any sulattice lattice isomorphc to $l^s$.