Coordinate systems and distributional embeddings in Bourgain-Rosenthal-Schechtman spaces: a framework for operator reduction

TL;DR

This paper develops a framework for operator reduction in Bourgain-Rosenthal-Schechtman spaces using coordinate systems and distributional embeddings, enabling approximation of operators by diagonal ones.

Contribution

It introduces an explicit unconditional finite-dimensional decomposition with strong properties, facilitating operator reduction and distributional representations in these complex spaces.

Findings

Constructed an explicit FDD with strong reproducing properties.

Proved that bounded operators can be approximated by scalar FDD-diagonal operators.

Established the factorization property for the MDS bases in the limit spaces.

Abstract

For every $1\leq \alpha<\omega_1$, we construct an explicit unconditional finite-dimensional decomposition (FDD) $(X_\lambda)_{\lambda\in\mathcal{T}_\alpha}$ of the Bourgain-Rosenthal-Schechtman space $R_\alpha^{p,0}$ by blocking its standard martingale difference sequence (MDS) basis. This FDD has strong reproducing properties and supports a theory of distributional representations between the spaces $R_\alpha^{p,0}$, $1\leq \alpha<\omega_1$. We use this framework to prove an approximate orthogonal reduction: every bounded linear operator on a limit space $R_\alpha^{p,0}$ is, via a distributional embedding and up to arbitrary precision, reduced to a scalar FDD-diagonal operator. As a consequence, the standard MDS bases of the limit spaces $R_\alpha^{p,0}$ satisfy the factorization property.