Lack of isomorphic embeddings of symmetric function spaces into operator ideals

TL;DR

This paper investigates conditions under which symmetric function spaces cannot be embedded into certain operator ideals, extending known results from classical $L_p$ and $\\ell_p$ spaces to more general symmetric spaces.

Contribution

It provides new criteria for non-embeddability of symmetric function spaces into symmetric ideals of compact operators, broadening previous specific cases.

Findings

Identifies conditions preventing embeddings of symmetric spaces into operator ideals.

Extends classical results from $L_p$ and $\\ell_p$ spaces to general symmetric spaces.

Provides criteria applicable to a wide class of symmetric function and sequence spaces.

Abstract

Let $E(0,1)$ be a symmetric space on $(0,1)$ and $C_F$ be a symmetric ideal of compact operators on the Hilbert space $\ell_2$ associated with a symmetric sequence space $F$. We give several criteria for $E(0,1)$ and $ F$ so that $E(0,1)$ does not embed into the ideal $C_F$, extending the result for the case when $E(0,1)=L_p(0,1)$ and $F=\ell_p $, $1\le p<\infty$, due to Arazy and Lindenstrauss.