DOI: 10.1515/ms-2021-0059 Math. Slovaca 71 (2021), No. 6, 1375–1400 ON THE FACTORABLE SPACES OF ABSOLUTELY p-SUMMABLE, NULL, CONVERGENT, AND BOUNDED SEQUENCES FEYZ ˙ I BAS ¸AR* , c — HADI ROOPAEI** (Communicated by Tomasz Natkaniec ) ABSTRACT. Let F denote the factorable matrix and X ∈{ℓp,c 0 , c, ℓ∞}. In this study, we introduce the domains X(F ) of the factorable matrix in the spaces X. Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces X(F ). We obtain the necessary and sufficient con- ditions on an infinite matrix belonging to the classes (ℓp(F ),ℓ∞), (ℓp(F ),f ) and (X, Y (F )) of matrix transformations, where Y denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix F and derive two factorizations for the Ces` aro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix F . Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix. c 2021 Mathematical Institute Slovak Academy of Sciences 1. Introduction In [45], Ng and Lee introduced the Ces` aro sequence spaces X p and X ∞ of non-absolute type as the domains of Ces` aro matrix C of order one in the spaces ℓ p and ℓ ∞ , where 1 ≤ p< ∞. Recently, S ¸eng¨ on¨ ul and Ba¸ sar [57] studied the Ces` aro sequence spaces  c 0 and  c of non-absolute type as the domains of Ces` aro matrix C of order one in the spaces c 0 and c, respectively. Besides, Ba¸ sar and Altay [6], and Altay and Ba¸ sar [1] introduced the space bv p as the domain of backward difference matrix Δ in the space ℓ p with 1 ≤ p ≤∞ and 0 <p< 1, repectively. Quite recently, Roopaei et al. [54] have investigated the Ces` aro sequence spaces ℓ p (C n ) and ℓ ∞ (C n ) as the domains of Ces` aro matrix C n of order n in the spaces ℓ p and ℓ ∞ , where 1 ≤ p< ∞. By using the domains of special triangle matrices in the classical sequence spaces, many authors introduced and studied new Banach spaces. For recent developments in this direction, we refer the reader to the articles in [7, 9–11, 22, 31, 33–35, 40, 47, 55] and textbooks/monographs [5, 8] and [38, 39], and the references therein. In this paper, as a natural continuation of Roopaei and Ba¸ sar [51], and S ¸eng¨ on¨ ul and Ba¸ sar [57], we introduce the sequence spaces ℓ p (F ), c 0 (F ), c(F ) and ℓ ∞ (F ) by the domains of factorable 2020 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 26D15, 46A45, 46B45; Secondary 40C05, 40G05, 47B37, 47B39. K e y w o r d s: domain of factorable matrix, almost convergence, weighted mean matrix, Hilbert matrix, gamma matrix, Ces`aro matrix. c Corresponding author. 1375 AUTHOR COPY