Embeddings between sequence variable Lebesgue spaces, strict and finitely strict singularity

Abstract For two variable Lebesgue spacesℓ p n $\ell _{p_n}$ andℓ q n $\ell _{q_n}$ , with0 < p n , q n < ∞ $0<p_n, q_n <$ , we provide necessary and sufficient conditions under which the natural embeddingsi d : ℓ p n → ℓ q n$id:\ell _{p_n} \ell _{q_n}$ are strictly or finitely strictly singular. We also provide estimates for the Bernstein numbers of the natural embeddingi d $id$ and show how they depend on the exponentsp n $p_n$ andq n $q_n$ .