Electronic Theses and Dissertations

Date of Award

2018

Document Type

Dissertation

Degree Name

Ph.D. in Mathematics

Department

Mathematics

First Advisor

Qingying Bu

Second Advisor

James Cizdziel

Third Advisor

Gerard Buskes

Abstract

For Banach lattices E1,…, Em and F with 1-unconditional bases, we show that the monomial sequence forms a 1-unconditional basis of Lr(E1,…, Em;F), the Banach lattice of all regular m-linear operators from E1×···× Em to F, if and only if each basis of E1,…,Em is shrinking and every positive m-linear operator from E 1×···×Em to F is weakly sequentially continuous. As a consequence, we obtain necessary and sufficient conditions for which the m-fold Fremlin projective tensor product E1⊗ |π|··· ⊗|π|E m (resp. the m-fold positive injective tensor product E1⊗|ϵ|··· ⊗ |ϵ|Em) has a shrinking basis or a boundedly complete basis. For Banach lattices E and F with 1-unconditional bases, we show that the monomial sequence forms a 1-unconditional basis of Pr, the Banach lattice of all regular m-homogeneous polynomials from E to F, if and only if E has a shrinking basis and every positive m-homogeneous polynomial from E to F is weakly sequentially continuous. As a consequence, we obtain necessary and sufficient conditions for which the m-fold symmetric positive projective tensor product ⊗m,s,|π|E (resp. the m-fold symmetric positive injective tensor product ⊗ m,s,|ϵ|E) has a shrinking basis or a boundedly complete basis. For a vector lattice E and n ∈ N, let ⊗n,sE denote the n-fold Fremlin vector lattice symmetric tensor product of E. For m,n ∈ N with m > n, we prove that (i) if ⊗ m,sE is uniformly complete then ⊗n,sE is positively isomorphic to a complemented subspace of ⊗ m,sE, and (ii) if there exists &phis; ∈ E∼+ such that ker(&phis;) is a projection band in E then ⊗n,sE is lattice isomorphic to a projection band of ⊗ m,sE. We also obtain analogous results for the n-fold Fremlin Banach lattice symmetric tensor product ⊗n,s,|π| E of E where E is a Banach lattice.

Included in

Mathematics Commons

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