Multipliers on Generalized Mixed Norm Sequence Spaces

Abstract

Given $\mathrm{1}\le p,q\le \mathrm{\infty }$ and sequences of integers $(n_k{)}_k$ and ${(n_k^{\prime })}_k$ such that $n_k\le n_k^{\prime }\le n_{k+\mathrm{1}}$, the generalized mixed norm space ${\ell }^{ℐ}(p,q)$ is defined as those sequences $(a_j{)}_j$ such that $(({\sum }_{j\in I_k}\mathrm{‍}|a_j{|}^p{)}^{\mathrm{1}/p}{)}_k\in {\ell }^q$ where . The necessary and sufficient conditions for a sequence $\lambda =({\lambda }_j{)}_j$ to belong to the space of multipliers $({\ell }^{ℐ}(r,s),{\ell }^{𝒥}(u,v))$, for different sequences $ℐ$ and $𝒥$ of intervals in ${\mathbb{N}}_{\mathrm{0}}$, are determined.