A+ CATEGORY SCIENTIFIC UNIT

Diagonals of projective tensor products and orthogonally additive polynomials

Volume 221 / 2014

Qingying Bu, Gerard Buskes Studia Mathematica 221 (2014), 101-115 MSC: 46G25, 47H60, 46B28. DOI: 10.4064/sm221-2-1

Abstract

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.

Authors

  • Qingying BuDepartment of Mathematics
    University of Mississippi
    University, MS 38677, U.S.A.
    e-mail
  • Gerard BuskesDepartment of Mathematics
    University of Mississippi
    University, MS 38677, U.S.A.
    e-mail

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