On σσ-conservative matrices and compact operators on the space VσVσ

The idea of σσ-conservative matrices was studied by Schaefer [P. Schaefer, Infinite matrices and invariant means, Proc. Amer. Math. Soc. 36 (1) (1972) 104–110]. In the present paper, we apply the Hausdorff measure of noncompactness to obtain the sufficient condition for a σσ-conservative matrix to define a compact operator. Further, we deduce the necessary and sufficient conditions for some matrix operators on the space VσVσ to be compact.