On almost -summable double sequence spaces.

The concept of a four-dimensional generalized difference matrix and its domain on some double sequence spaces was recently introduced and studied by Tuğ and Başar (AIP Conference Proceedings, vol. 1759, 2016) and Tuğ (J. Inequal. Appl. 2017(1):149, 2017). In this present paper, as a natural continuation of (J. Inequal. Appl. 2017(1):149, 2017), we introduce new almost null and almost convergent double sequence spaces [Formula: see text] and [Formula: see text] as the four-dimensional generalized difference matrix [Formula: see text] domain in the spaces [Formula: see text] and [Formula: see text], respectively. Firstly, we prove that the spaces [Formula: see text] and [Formula: see text] of double sequences are Banach spaces under some certain conditions. Then we give an inclusion relation of these new almost convergent double sequence spaces. Moreover, we identify the -dual, [Formula: see text]-dual and -dual of the space [Formula: see text]. Finally, we characterize some new matrix classes [Formula: see text], [Formula: see text], and we complete this work with some significant results.