On sign embeddings and narrow operators on $L_2$

TL;DR

This paper reviews weak and sign embeddings, discusses narrow operators in Banach space geometry, and proves that Rosenthal's characterization of narrow operators on L1 extends to L2, answering a longstanding open question.

Contribution

It extends Rosenthal's characterization of narrow operators from L1 to L2, providing new insights into operator theory on Hilbert spaces.

Findings

Rosenthal's characterization holds for L2 operators

Examples of applications in Banach space geometry

Open problems for 1<p<2 cases

Abstract

The goal of this note is two-fold. First we present a brief overview of "weak" embeddings, with a special emphasis on sign embeddings which were introduced by H. P. Rosenthal in the early 1980s. We also discuss the related notion of narrow operators, which was introduced by A. Plichko and M. Popov in 1990. We give examples of applications of these notions in the geometry of Banach spaces and in other areas of analysis. We also present some open problems. In the second part we prove that Rosenthal's celebrated characterization of narrow operators on $L_1$ is also true for operators on $L_2$. This answers, for $p=2$, a question posed by Plichko and Popov in 1990. For $1<p<2$ the problem remains open.