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

Konstantinos Konstantos; Pavlos Motakis

Paper

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

Abstract

For every

, we construct an explicit unconditional finite-dimensional decomposition (FDD)

of the Bourgain-Rosenthal-Schechtman space

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

. We use this framework to prove an approximate orthogonal reduction: every bounded linear operator on a limit space

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

satisfy the factorization property.