Two methods for the characterization of compact operators between BK spaces

Abstract

We establish some identities or inequalities for the Hausdorff measure of noncompactness for operators $L\in\mathcal{B}(X,Y)$ when $X=\ell_{p}$ $(p \in [1, \infty))$ and $Y=c$; $X=\ell_{p}$ $(p \in (1, \infty))$ and $Y=\ell_{\infty}$; $X=bv_{0}$ and $Y=c$; $X=c_{0}(\Delta),c(\Delta),\ell_{\infty}(\Delta)$ and $Y=\ell_{\infty}$. These identities and estimates are used to establish necessary and sufficient conditions for the operators to be compact. Furthermore, we apply a result by Sargent to establish necessary and sufficient conditions for operators in $\mathcal{B}(bv_{0},\ell_{\infty})$ and $\mathcal{B}(\ell_{1},Y)$ to be compact, where $Y=w_{\infty},v_{\infty}, [c]_{\infty}$.