Criteria for the boundedness of a certain class of matrix operators from lpv into lqu

Abstract

One of the main aims in the theory of matrices is to find necessary and sufficient conditions for the elements of any matrix so that the corresponding matrix operator maps continuously from one normed space into another one. Thus, it is very important to find the norm of the matrix operator, at least, to find upper and lower estimates of it. This problem in Lebesgue spaces of sequences in the general case is still open. This paper deals with the problem of boundedness of matrix operators from lpv into lqu for 1 < q < p < ∞, and we obtain necessary and sufficient conditions of this problem when matrix operators belong to the classes O ± 2 satisfying weaker conditions than Oinarov’s condition.