Generalized Almost Convergence and Core Theorems of Double Sequences

Abstract

The idea of $[\lambda , \mu ]$-almost convergence (briefly, ${\mathcal{F}}_{[\lambda , \mu ]}$-convergence) has been recently introduced and studied by Mohiuddine and Alotaibi (2014). In this paper first we define a norm on ${\mathcal{F}}_{[\lambda , \mu ]}$ such that it is a Banach space and then we define and characterize those four-dimensional matrices which transform ${\mathcal{F}}_{[\lambda , \mu ]}$-convergence of double sequences $x=(x_{jk})$ into ${\mathcal{F}}_{[\lambda , \mu ]}$-convergence. We also define a ${\mathcal{F}}_{[\lambda , \mu ]}$-core of $x=(x_{jk})$ and determine a Tauberian condition for core inclusions and core equivalence.