Boundedness Problem of Four-Dimensional Matrices on the Domain Spaces of $$\mathcal {L}_p$$ L p

Abstract

In this paper, we introduce and study the domain of an arbitrary four-dimensional summability matrix in the double sequence space $$\mathcal {L}_p$$ and focus on the boundedness problem of four-dimensional operators on it. We consider the class of four-dimensional Hausdorff matrices as operators mapping $$\mathcal {L}_p$$ into these domains and provide a Hardy type formula for their operator norms and lower bounds. Then we apply our results to some special domains of $$\mathcal {L}_p$$ such as the double Laurent, Taylor and Euler sequence spaces. Finally, we provide a general upper estimate for the operator norms of four-dimensional matrices which have real and complex entries with certain conditions.