A version of Kalton's theorem for the space of regular homogeneous polynomials on Banach lattices (Q6630627)

Given Banach spaces \(E\) and \(F\), a continuous function \(P\colon E\to F\) is said to be an \(m\)-homogeneous polynomial if there is a continuous \(m\)-linear mapping \(A_P\colon E \times\cdots\times E\to F\) such that \(P(x)=A_P(x,\ldots,x)\) for all \(x\) in \(E\). When \(E\) and \(F\) are Banach lattices, an \(m\)-homogenous \(P\) is said to be positive if \(A_P(x_1, \ldots, x_m)\ge 0\) whenever \(x_1,\ldots,x_m\ge 0\). An \(m\)-homogeneous polynomial \(P\) is said to be regular if it is the difference of two positive polynomials. The space of all regular \(m\)-homogeneous polynomials from \(E\) to \(F\) is denoted by \(\mathcal{P}^r(^mE,F)\), while the space of all regular \(m\)-homogeneous polynomials which are weakly continuous on bounded sets is denoted by \(\mathcal{P} ^r_w(^mE,F)\). The space \(\mathcal{P}^r(^mE,\mathbb{R})\) is a dual space, being the dual of \({\widehat\otimes}_{m,s,|\pi|}E\), the \(m\)-fold symmetric tensor product of \(E\) endowed with the positive projective norms. Given an \(m\)-homogeneous polynomial \(P\colon E\to F\), \(\tilde P\) denotes the (canonical) Aron-Berner extension of \(P\) to a polynomial from \(E''\) (the bidual of \(E\)) to \(F''\) (the bidual of \(F\)). The main result of the paper is a version of Kalton's Theorem on weak convergence of compact operators for weakly continuous regular polynomials. Specifically, if \(E\) is a Banach lattice and \((P_k)_k\) is a sequence in \(\mathcal{P}^r_w(^mE,\mathbb{R})\), with \(P_k\ge 0\) all \(k\), then \(P_k\to 0\) weakly in \(\mathcal{P}^r_w(^mE,\mathbb{R})\) if and only if \({\tilde P_k}(z)\to 0\) for every \(z\) in \(E''\).\N\NA Banach lattice is said to have the PGP if every positive weak\({}^*\) null sequence in \(E'\) is weakly null and is said to have the RAP if for every compact subset \(C\) of \(E\) and any \(\epsilon>0\), there exists a finite rank operator \(T_{C,\epsilon}\) on \(E\) such that \(\||T_{C,\epsilon} - id|(x)\|\le \epsilon\) for all \(x\in C\). When the \(T_{C,\epsilon}\) can be chosen so that \(\|T_{C,\epsilon}\|\) are bounded \(E\) is said to have the BRAP. Using the above extension of Kalton's Theorem the author shows that if \(E\) has the PGP and every positive \(m\)-homogeneous polynomial from \(E\) to \(\mathbb{R}\) is weakly continuous on bounded sets then \({\widehat\otimes}_{m,s,|\pi|}E\) has the PGP. The converse is shown to be true when \(E'\) has the BRAP. When \(E\) and \(F\) are reflexive Banach lattices, the extension of Kalton's Theorem is used to show that if every \(m\)-homogeneous polynomial from \(E\) to \(F\) is weakly sequentially continuous then \(\mathcal{P}^r(^m E,F)\) is reflexive. Moreover, if \(E'\) and \(F\) have the BRAP then the converse is also true. Examples of pairs of Banach lattices \(E\) and \(F\) so that \(\mathcal{P}^r (^mE,F)\) is reflexive are provided.