Two Remarks on Primary Spaces

TL;DR

This paper investigates the factorization properties of operators on certain Banach spaces, proving that the identity operator factors through any operator or its complement, and offers combinatorial proofs replacing previous reliance on Szemeredi's theorem.

Contribution

It extends factorization results to spaces (H^p(T)) for 1<p<, using combinatorics of dyadic intervals instead of Szemeredi's theorem.

Findings

Identity factors through any operator or its complement on (H^1(T)).

Analogous results are established for (H^p(T))) spaces for 1<p<.

Combinatorial methods replace the dependence on Szemeredi's theorem.

Abstract

We prove that for any operator $T$ on $ \ell^\infty(H^1 (\bT))$, the identity factores through $T$ or $\Id - T$. We re-prove analogous results of H.M. Wark for the spaces $\ell^infty(H^p(\bT))$, $1<p <\infty$. In the present paper direct combinatorics of colored dyadic intervals replaces the dependence on Szemeredi's theorem in the work of H. M. Wark.