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.