Lack of isomorphic embedding from Lorentz function spaces into Lorentz operator ideals

Abstract

In the present paper, we study the existence of isomorphic embeddings from Lorentz function spaces 𝐿 𝑝 1 , 𝑞 1 into Lorentz ideals 𝒞 𝑝 2 , 𝑞 2 in 𝐵 ⁡ ( 𝐻 ) . In particular, we show that, for 𝑝 , 𝑞 ∈ ( 0 , ∞ ) , the quasi-Banach function space 𝐿 𝑝 , 𝑞 ⁡ ( 0 , 1 ) isomorphically embeds into the ideal 𝒞 𝑝 , 𝑞 if and only if ( 𝑝 , 𝑞 ) = ( 2 , 2 ) . This extends several existing results in the literature and answers a question due to Astashkin et al. [6] .