Some Properties of Orthogonally Additive Homogeneous Polynomials on Banach Lattices

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 \(l^s\).

For citation: Kusraeva, Z. A. and Siukaev, S. N. Some Properties of Orthogonally Additive Homogeneous Polynomials on Banach Lattices, Vladikavkaz Math. J., 2020, vol. 22, no. 4, pp. 92-103 (in Russian). DOI 10.46698/d4799-1202-6732-b

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