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 ∈ B(X, Y ) when X = p (1 ≤ p < ∞) and Y = c; X = p (1 < p < ∞) and Y = ∞ ; X = bv 0 and Y = c; X = c 0 (∆), c(∆), ∞ (∆) and Y = ∞ . 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 B(bv 0 , ∞ ) and B( 1 , Y ) to be compact, where Y = w ∞ , v ∞ , [c] ∞ .