{"id":{"repo_id":"york","oai_identifier":"oai:yorkspace.library.yorku.ca:10315/43955"},"canonical_url":"https://search.dev.ndltd.org/etd/york/oai:yorkspace.library.yorku.ca:10315/43955","repository":{"repo_id":"york","name":"York University","base_url":"https://yorkspace.library.yorku.ca/oai/request"},"display":{"title":"Orthogonal Factors of Operators on the Rosenthal Spaces and the Bourgain-Rosenthal-Schechtman Spaces","abstract":"We study factorization properties of bounded linear operators on the Rosenthal spaces $X_{p,w}$ and the Bourgain-Rosenthal-Schechtman spaces $R_{\\alpha}^{p,0}$, $1 \\leq \\alpha < \\omega_{1}$. Specifically, we prove that the Rosenthal spaces $X_{p,w}$ and the limit Bourgain-Rosenthal-Schechtman spaces $R_{\\alpha}^{p,0}$, equipped with their natural bases, have the factorization property, and that the isomorphic spaces $R_{\\omega}^{p,0}$ and $X_{p,w}$ have the primary factorization property. We establish the factorization property of the spaces $R_{\\omega}^{p,0}$ and $X_{p,w}$ with their respective bases separately. For the spaces $X_{p,w}$, the proof relies on the notion of a strategically reproducible basis. In contrast, for $R_{\\omega}^{p,0}$, the proof is based on an approximate orthogonal reduction to a diagonal operator: every bounded linear operator on $R_{\\omega}^{p,0}$ can, via distributional embedding and up to arbitrary precision, be reduced to a diagonal operator. Moreover, we show that $R_{\\omega}^{p,0}$ has the primary factorization property, which immediately implies the same property for $X_{p,w}$. The proof of the aforementioned reduction relies on an approximate orthogonal reduction to a scalar multiple of the identity operator. In addition, we establish the factorization property for the limit spaces $R_{\\alpha}^{p,0}$ with their standard martingale difference sequence bases. Although this proof is derived from an approximate orthogonal reduction to a scalar finite-dimensional decomposition (FDD)-diagonal operator, the construction is significantly more involved than in the case $\\alpha = \\omega$. For every $1 \\leq \\alpha < \\omega_1$, we construct an explicit unconditional FDD $(X_\\lambda)_{\\lambda \\in \\mathcal{T}_\\alpha}$ of the Bourgain-Rosenthal-Schechtman space $R_\\alpha^{p,0}$. This FDD has strong reproducing properties and supports a theory of distributional representations between the spaces $R_\\alpha^{p,0}$, for $1 \\leq \\alpha < \\omega_1$. We use this framework to establish the approximate orthogonal reduction to a scalar FDD-diagonal operator.","abstract_html":"We study factorization properties of bounded linear operators on the Rosenthal spaces <span class=\"etd-inline-math\">X<sub>p,w</sub></span> and the Bourgain-Rosenthal-Schechtman spaces <span class=\"etd-inline-math\">R<sub>&alpha;</sub><sup>p,0</sup></span>, <span class=\"etd-inline-math\">1 \\leq &alpha; &lt; &omega;<sub>1</sub></span>. Specifically, we prove that the Rosenthal spaces <span class=\"etd-inline-math\">X<sub>p,w</sub></span> and the limit Bourgain-Rosenthal-Schechtman spaces <span class=\"etd-inline-math\">R<sub>&alpha;</sub><sup>p,0</sup></span>, equipped with their natural bases, have the factorization property, and that the isomorphic spaces <span class=\"etd-inline-math\">R<sub>&omega;</sub><sup>p,0</sup></span> and <span class=\"etd-inline-math\">X<sub>p,w</sub></span> have the primary factorization property. We establish the factorization property of the spaces <span class=\"etd-inline-math\">R<sub>&omega;</sub><sup>p,0</sup></span> and <span class=\"etd-inline-math\">X<sub>p,w</sub></span> with their respective bases separately. For the spaces <span class=\"etd-inline-math\">X<sub>p,w</sub></span>, the proof relies on the notion of a strategically reproducible basis. In contrast, for <span class=\"etd-inline-math\">R<sub>&omega;</sub><sup>p,0</sup></span>, the proof is based on an approximate orthogonal reduction to a diagonal operator: every bounded linear operator on <span class=\"etd-inline-math\">R<sub>&omega;</sub><sup>p,0</sup></span> can, via distributional embedding and up to arbitrary precision, be reduced to a diagonal operator. Moreover, we show that <span class=\"etd-inline-math\">R<sub>&omega;</sub><sup>p,0</sup></span> has the primary factorization property, which immediately implies the same property for <span class=\"etd-inline-math\">X<sub>p,w</sub></span>. The proof of the aforementioned reduction relies on an approximate orthogonal reduction to a scalar multiple of the identity operator. In addition, we establish the factorization property for the limit spaces <span class=\"etd-inline-math\">R<sub>&alpha;</sub><sup>p,0</sup></span> with their standard martingale difference sequence bases. Although this proof is derived from an approximate orthogonal reduction to a scalar finite-dimensional decomposition (FDD)-diagonal operator, the construction is significantly more involved than in the case <span class=\"etd-inline-math\">&alpha; = &omega;</span>. For every <span class=\"etd-inline-math\">1 \\leq &alpha; &lt; &omega;<sub>1</sub></span>, we construct an explicit unconditional FDD <span class=\"etd-inline-math\">(X<sub>\\</sub>lambda)<sub>\\lambda \\in \\mathcal{T}<sub>\\</sub>alpha</sub></span> of the Bourgain-Rosenthal-Schechtman space <span class=\"etd-inline-math\">R<sub>\\</sub>alpha<sup>p,0</sup></span>. This FDD has strong reproducing properties and supports a theory of distributional representations between the spaces <span class=\"etd-inline-math\">R<sub>\\</sub>alpha<sup>p,0</sup></span>, for <span class=\"etd-inline-math\">1 \\leq &alpha; &lt; &omega;<sub>1</sub></span>. We use this framework to establish the approximate orthogonal reduction to a scalar FDD-diagonal operator.","abstract_has_math":true,"creators":["Konstantos, Konstantinos"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Motakis, Pavlos"],"committee_chairs":[],"committee_members":[],"year":2026,"date_issued":"2026-07-24","date_published":"2026-07-24","updated_at":"2026-08-21T16:51:02Z","subjects":["Mathematics"],"languages":["en"],"rights":["Author owns copyright, except where explicitly noted. Please contact the author directly with licensing requests."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/10315/43955","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"source_record":{"url":"https://yorkspace.library.yorku.ca/oai/request?verb=GetRecord&metadataPrefix=dim&identifier=oai%3Ayorkspace.library.yorku.ca%3A10315%2F43955","prefix":"dim"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Motakis, Pavlos"]},{"key":"dc:creator","label":"Author","values":["Konstantos, Konstantinos"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2026-07-24T15:47:26Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2026-07-24T15:47:26Z"]},{"key":"dc:date.issued","label":"Date","values":["2026-07-24"]},{"key":"dc:type","label":"Dc Type","values":["Electronic Thesis or Dissertation"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Author owns copyright, except where explicitly noted. Please contact the author directly with licensing requests."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10315/43955"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We study factorization properties of bounded linear operators on the Rosenthal spaces $X_{p,w}$ and the Bourgain-Rosenthal-Schechtman spaces $R_{\\alpha}^{p,0}$, $1 \\leq \\alpha < \\omega_{1}$. Specifically, we prove that the Rosenthal spaces $X_{p,w}$ and the limit Bourgain-Rosenthal-Schechtman spaces $R_{\\alpha}^{p,0}$, equipped with their natural bases, have the factorization property, and that the isomorphic spaces $R_{\\omega}^{p,0}$ and $X_{p,w}$ have the primary factorization property. We establish the factorization property of the spaces $R_{\\omega}^{p,0}$ and $X_{p,w}$ with their respective bases separately. For the spaces $X_{p,w}$, the proof relies on the notion of a strategically reproducible basis. In contrast, for $R_{\\omega}^{p,0}$, the proof is based on an approximate orthogonal reduction to a diagonal operator: every bounded linear operator on $R_{\\omega}^{p,0}$ can, via distributional embedding and up to arbitrary precision, be reduced to a diagonal operator. Moreover, we show that $R_{\\omega}^{p,0}$ has the primary factorization property, which immediately implies the same property for $X_{p,w}$. The proof of the aforementioned reduction relies on an approximate orthogonal reduction to a scalar multiple of the identity operator. In addition, we establish the factorization property for the limit spaces $R_{\\alpha}^{p,0}$ with their standard martingale difference sequence bases. Although this proof is derived from an approximate orthogonal reduction to a scalar finite-dimensional decomposition (FDD)-diagonal operator, the construction is significantly more involved than in the case $\\alpha = \\omega$. For every $1 \\leq \\alpha < \\omega_1$, we construct an explicit unconditional FDD $(X_\\lambda)_{\\lambda \\in \\mathcal{T}_\\alpha}$ of the Bourgain-Rosenthal-Schechtman space $R_\\alpha^{p,0}$. This FDD has strong reproducing properties and supports a theory of distributional representations between the spaces $R_\\alpha^{p,0}$, for $1 \\leq \\alpha < \\omega_1$. We use this framework to establish the approximate orthogonal reduction to a scalar FDD-diagonal operator."]},{"key":"dc:title","label":"Title","values":["Orthogonal Factors of Operators on the Rosenthal Spaces and the Bourgain-Rosenthal-Schechtman Spaces"]}]}],"canonical_facts":{"dc:contributor.advisor":["Motakis, Pavlos"],"dc:creator":["Konstantos, Konstantinos"],"dc:date.accessioned":["2026-07-24T15:47:26Z"],"dc:date.available":["2026-07-24T15:47:26Z"],"dc:date.issued":["2026-07-24"],"dc:description.abstract":["We study factorization properties of bounded linear operators on the Rosenthal spaces $X_{p,w}$ and the Bourgain-Rosenthal-Schechtman spaces $R_{\\alpha}^{p,0}$, $1 \\leq \\alpha < \\omega_{1}$. Specifically, we prove that the Rosenthal spaces $X_{p,w}$ and the limit Bourgain-Rosenthal-Schechtman spaces $R_{\\alpha}^{p,0}$, equipped with their natural bases, have the factorization property, and that the isomorphic spaces $R_{\\omega}^{p,0}$ and $X_{p,w}$ have the primary factorization property. We establish the factorization property of the spaces $R_{\\omega}^{p,0}$ and $X_{p,w}$ with their respective bases separately. For the spaces $X_{p,w}$, the proof relies on the notion of a strategically reproducible basis. In contrast, for $R_{\\omega}^{p,0}$, the proof is based on an approximate orthogonal reduction to a diagonal operator: every bounded linear operator on $R_{\\omega}^{p,0}$ can, via distributional embedding and up to arbitrary precision, be reduced to a diagonal operator. Moreover, we show that $R_{\\omega}^{p,0}$ has the primary factorization property, which immediately implies the same property for $X_{p,w}$. The proof of the aforementioned reduction relies on an approximate orthogonal reduction to a scalar multiple of the identity operator. In addition, we establish the factorization property for the limit spaces $R_{\\alpha}^{p,0}$ with their standard martingale difference sequence bases. Although this proof is derived from an approximate orthogonal reduction to a scalar finite-dimensional decomposition (FDD)-diagonal operator, the construction is significantly more involved than in the case $\\alpha = \\omega$. For every $1 \\leq \\alpha < \\omega_1$, we construct an explicit unconditional FDD $(X_\\lambda)_{\\lambda \\in \\mathcal{T}_\\alpha}$ of the Bourgain-Rosenthal-Schechtman space $R_\\alpha^{p,0}$. This FDD has strong reproducing properties and supports a theory of distributional representations between the spaces $R_\\alpha^{p,0}$, for $1 \\leq \\alpha < \\omega_1$. We use this framework to establish the approximate orthogonal reduction to a scalar FDD-diagonal operator."],"dc:identifier.uri":["https://hdl.handle.net/10315/43955"],"dc:language":["en"],"dc:rights":["Author owns copyright, except where explicitly noted. Please contact the author directly with licensing requests."],"dc:subject":["Mathematics"],"dc:title":["Orthogonal Factors of Operators on the Rosenthal Spaces and the Bourgain-Rosenthal-Schechtman Spaces"],"dc:type":["Electronic Thesis or Dissertation"]},"updated_at":"2026-08-21T16:51:02Z"}