Orthogonal Factors of Operators on the Rosenthal Spaces and the Bourgain-Rosenthal-Schechtman Spaces

We study factorization properties of bounded linear operators on the Rosenthal spaces X_p,w and the Bourgain-Rosenthal-Schechtman spaces R_α^p,0, 1 \leq α < ω₁. Specifically, we prove that the Rosenthal spaces X_p,w and the limit Bourgain-Rosenthal-Schechtman spaces R_α^p,0, equipped with their natural bases, have the factorization property, and that the isomorphic spaces R_ω^p,0 and X_p,w have the primary factorization property. We establish the factorization property of the spaces R_ω^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_ω^p,0, the proof is based on an approximate orthogonal reduction to a diagonal operator: every bounded linear operator on R_ω^p,0 can, via distributional embedding and up to arbitrary precision, be reduced to a diagonal operator. Moreover, we show that R_ω^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_α^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 α = ω. For every 1 \leq α < ω₁, 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 α < ω₁. We use this framework to establish the approximate orthogonal reduction to a scalar FDD-diagonal operator.

Author dc:creator

• Konstantos, Konstantinos

Advisor dc:contributor.advisor

• Motakis, Pavlos

• Mathematics

Statement dc:rights

• Author owns copyright, except where explicitly noted. Please contact the author directly with licensing requests.

Language dc:language

en

Handle dc:identifier.uri

https://hdl.handle.net/10315/43955

OAI record oai:identifier

https://yorkspace.library.yorku.ca/oai/request?verb=GetRecord&metadataPrefix=dim&identifier=oai%3Ayorkspace.library.yorku.ca%3A10315%2F43955