Diagonals of projective tensor products and orthogonally additive polynomials

Qingying Bu¹ ; Gerard Buskes¹

¹ Department of Mathematics University of Mississippi University, MS 38677, U.S.A.

Studia Mathematica, Tome 221 (2014) no. 2, pp. 101-115

Résumé

Let $E$ be a Banach space with $1$-unconditional basis. Denote by $\varDelta(\mathop{\hat{\otimes}_{n,\pi}}E)\!$ (resp. $\varDelta(\mathop{\hat{\otimes}_{n,s,\pi}}E)$) the main diagonal space of the $n$-fold full (resp. symmetric) projective Banach space tensor product, and denote by $\varDelta(\mathop{\hat{\otimes}_{n,|\pi|}}E)$ (resp. $\varDelta(\mathop{\hat{\otimes}_{n,s,|\pi|}}E)$) the main diagonal space of the $n$-fold full (resp. symmetric) projective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic, and in addition, that they are isometrically lattice isomorphic to $E_{[n]}$, the completion of the $n$-concavification of $E$. Using these isometries, we also show that the norm of any (vector valued) continuous orthogonally additive homogeneous polynomial on $E$ equals the norm of its associated symmetric linear operator.

Détail

DOI : 10.4064/sm221-2-1

Keywords: banach space unconditional basis denote vardelta mathop hat otimes resp vardelta mathop hat otimes main diagonal space n fold full resp symmetric projective banach space tensor product denote vardelta mathop hat otimes resp vardelta mathop hat otimes main diagonal space n fold full resp symmetric projective banach lattice tensor product these main diagonal spaces pairwise isometrically isomorphic addition isometrically lattice isomorphic completion n concavification using these isometries norm vector valued continuous orthogonally additive homogeneous polynomial equals norm its associated symmetric linear operator