{"id":{"repo_id":"mississippi","oai_identifier":"oai:egrove.olemiss.edu:etd-2066"},"canonical_url":"https://search.dev.ndltd.org/etd/mississippi/oai:egrove.olemiss.edu:etd-2066","repository":{"repo_id":"mississippi","name":"University of Mississippi","base_url":"https://egrove.olemiss.edu/do/oai/"},"display":{"title":"Bases In Spaces Of Regular Multilinear Operators And Homogeneous Polynomials On Banach Lattices","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Navoyan, Khazhak Varazdat"],"institution":null,"degree_name":"Ph.D. in Mathematics","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Qingying Bu","James Cizdziel","Gerard Buskes"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-01-01T08:00:00Z","date_published":"2018-01-01T08:00:00Z","updated_at":"2026-07-24T03:06:14Z","subjects":["Complementation","Fremlin Tensor Product","Monomial Bases","Positive Tensor Products","Projection Band","Regular Multilinear Operators","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://egrove.olemiss.edu/etd/1067","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Qingying Bu","James Cizdziel","Gerard Buskes"]},{"key":"dc:creator","label":"Author","values":["Navoyan, Khazhak Varazdat"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2019-06-20T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D. in Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Complementation","Fremlin Tensor Product","Monomial Bases","Positive Tensor Products","Projection Band","Regular Multilinear Operators","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://egrove.olemiss.edu/etd/1067"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:title","label":"Title","values":["Bases In Spaces Of Regular Multilinear Operators And Homogeneous Polynomials On Banach Lattices"]}]}],"canonical_facts":{"dc:contributor":["Qingying Bu","James Cizdziel","Gerard Buskes"],"dc:creator":["Navoyan, Khazhak Varazdat"],"dc:date.available":["2019-06-20T07:00:00Z"],"dc:description.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."],"dc:identifier":["https://egrove.olemiss.edu/etd/1067"],"dc:subject":["Complementation","Fremlin Tensor Product","Monomial Bases","Positive Tensor Products","Projection Band","Regular Multilinear Operators","Mathematics"],"dc:title":["Bases In Spaces Of Regular Multilinear Operators And Homogeneous Polynomials On Banach Lattices"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D. in Mathematics"]},"updated_at":"2026-07-24T03:06:14Z"}