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 Xp,w and the Bourgain-Rosenthal-Schechtman spaces Rαp,0, 1≤α<ω1. Specifically, we prove that the Rosenthal spaces Xp,w and the limit Bourgain-Rosenthal-Schechtman spaces Rαp,0, equipped with their natural bases, have the factorization property, and that the isomorphic spaces Rωp,0 and Xp,w have the primary factorization property. We establish the factorization property of the spaces Rωp,0 and Xp,w with their respective bases separately. For the spaces Xp,w, the proof relies on the notion of a strategically reproducible basis. In contrast, for Rωp,0, the proof is based on an approximate orthogonal reduction to a diagonal operator: every bounded linear operator on Rωp,0 can, via distributional embedding and up to arbitrary precision, be reduced to a diagonal operator. Moreover, we show that Rωp,0 has the primary factorization property, which immediately implies the same property for Xp,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 Rαp,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≤α<ω1, we construct an explicit unconditional FDD (Xλ)λ∈Tα of the Bourgain-Rosenthal-Schechtman space Rαp,0. This FDD has strong reproducing properties and supports a theory of distributional representations between the spaces Rαp,0, for 1≤α<ω1. We use this framework to establish the approximate orthogonal reduction to a scalar FDD-diagonal operator.