On Arazy's problem concerning isomorphic embeddings of ideals of compact operators

Abstract

The primary objective of this paper is to study the problem concerning isomorphic embeddings of ideals of compact operators due to Arazy (see [5, Problem C] and [6, Problem (B)]), which is closely related to Pełczyński's problem of uniqueness of symmetric structure in such ideals. In particular, we show that for a wide class of symmetric sequence spaces E, if CF↪SE, then F=ℓ2 and E has a subspace isomorphic to ℓ2, where CX is the symmetric ideal of compact operators on the Hilbert space ℓ2 induced by a symmetric sequence space X and SX=(⊕n=1∞Mn)CX. This extends results due to Arazy [5,6]. As a consequence, we contribute to the Arazy–Pełczyński problem on the uniqueness of symmetric structures [5,6]. In particular, we can deal with the case when E has a subspace isomorphic to ℓ2 or c0, which was left untreated in [5,6].