Factorization of the identity through operators with large diagonal

Abstract

Given a Banach space~$X$ with an unconditional basis, we consider the following question: does the identity on~$X$ factor through every operator on~$X$ with large diagonal relative to the unconditional basis? We show that on Gowers' unconditional Banach space, there exists an operator for which the answer to the question is negative. By contrast, for any operator on the mixed-norm Hardy spaces $H^p(H^q)$, where $1\le p,q<\infty$, with the bi-parameter Haar system, this problem always has a positive solution. The spaces $L^p,1<p<\infty$, were treated first by Andrew~[{\em Studia Math.}~1979].

30 pages, 13 figures

References in corpus (3)

• Factorization in mixed norm Hardy and BMO spaces
• Localization and projections on bi-parameter BMO
• Factorization in $S{L}^{\infty }$

Cited by in corpus (5)

• Strategically reproducible bases and the factorization property
• Factorization in $S{L}^{\infty }$
• The factorization property of ${\ell }^{\infty }(X_k)$
• Continuous factorization of the identity matrix
• Orthogonal Factors of Operators on the Rosenthal $X_{p,w}$ spaces and the Bourgain-Rosenthal-Schechtman $R_{\omega }^p$ space