#### Title

Bases In Spaces Of Regular Multilinear Operators And Homogeneous Polynomials On Banach Lattices

#### 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

#### Relational Format

dissertation/thesis

#### 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.

#### Recommended Citation

Navoyan, Khazhak Varazdat, "Bases In Spaces Of Regular Multilinear Operators And Homogeneous Polynomials On Banach Lattices" (2018). *Electronic Theses and Dissertations*. 1067.

https://egrove.olemiss.edu/etd/1067