In 1981, Bourgain, Rosenthal and Schechtman constructed the $R_{\alpha}^{p}$ spaces, $\alpha < \omega_{1}$, yielding uncountably many pairwise non-isomorphic complemented subspaces of $L_{p}$, $1< p < \infty$. Each of them has a Schauder basis given by a 3-valued symmetric martingale difference sequence. In this work, we study factorization properties of operators on the $R_{\alpha}^{p}$ spaces. We say that a bounded linear operator $T$ on a Banach space $X$ is a factor of the identity operator $id$ on $X$ if there exist bounded linear operators $L,R \colon X \to X$ such that $id = LTR$. Factors of the identity have played a crucial role in Banach space theory, for example, in the study of closed ideals of the Banach algebra $\mathcal{L}(X)$ of all bounded linear operators on a Banach space $X$, and in the study of decompositions of classical Banach spaces (primary spaces). This motivates the following definition. We say that a Banach space $X$ with its Schauder basis $(e_{n})_{n \in \mathbb{N}}$ has the factorization property if every bounded linear operator $T \colon X \to X$ with large diagonal, i.e., such that $\inf_{n \in \mathbb{N}} \vert e_{n}^{\ast}(T(e_{n})) \vert \geq \delta > 0$, is a factor of the identity. In [Konstantos and Motakis, 2025a], we prove that, for $1< p < \infty$, the space $R_{\omega}^{p}$ with its natural basis has the factorization property. Additionally, in [Konstantos and Motakis, 2025b], we show that, for $1 < p < \infty$, the limit spaces $R_{\alpha}^{p}$ with their standard bases have the factorization property.

This is joint work with Pavlos Motakis.

\noindent\textbf{References}
J. Bourgain, H. P. Rosenthal, and G. Schechtman (1981), \emph{An ordinal $L_{p}$-index for Banach spaces, with application to complemented subspaces of $L_p$}, Ann. of Math. (2) 114(2):193–228.

[2] K. Konstantos and P. Motakis (2025a), \emph{Orthogonal factors of operators on the Rosenthal $X_{p,w}$ spaces and the Bourgain-Rosenthal-Schechtman $R_{\omega}^{p}$ space}, Journal of Functional Analysis, 288(5):110802.

[3] K. Konstantos and P. Motakis (2025b), \emph{Coordinate systems and distributional embeddings in Bourgain-Rosenthal-Schechtman spaces: a framework for operator reduction}.