Some matrix transformations and generalized core of double sequences

Abstract

A four-dimensional matrix transformation is said to be regular if it maps every bounded-convergent double sequence into a convergent sequence with the same limit. Firstly, Robison [G.M. Robison, Divergent double sequences and series, Trans. Amer. Math. Soc. 28 (1926) 50–73] presented the necessary and sufficient conditions for regular matrix transformations of double sequences. In this paper, the conditions of Robison are extended to the class of regular matrix transformations between the double sequence spaces c 2 P B ( p ) and c 2 P B . We also characterize the matrix classes ( 0 c 2 P B ( p ) , 0 c 2 P ( q ) ) , ( 0 c 2 P B ( p ) , 0 c 2 P B ( q ) ) , ( c 2 P B ( p ) , 0 c 2 P ( q ) ) , ( ℓ 2 ∞ ( p ) , 0 c 2 P ( q ) ) and ( ℓ 2 ∞ ( p ) , 0 c 2 P B ( q ) ) . Furthermore, we define the core of a real sequence belonging to the more general class ℓ 2 ∞ ( p ) and establish some results related to this new type of core by using our matrix classes ( c 2 P B ( p ) , c 2 P B ) and ( c 2 P B ( p ) , c 2 P B ) reg .