On the generalized Cesàro sequence spaces and some of their properties

Let Ct = (cnm)n,m∈ℕ denote the generalized Cesàro matrix defined by cnm = tn−m/n for m ≤ n and t ∈ [0, 1], and cnm = 0 otherwise. For each q ∈ [1, ∞), the associated generalized Cesàro sequence spaces cesqt are introduced. These spaces form Banach lattices under the coordinatewise order, equipped with a naturally defined order continuous norm. We prove that cesqt = cesq0 for all t ∈ [0, 1) and q ∈ [1, ∞), implying that the space cesqt is a weighted ℓq space with equivalent norm. Furthermore, we examine the inclusion relationships between cesq and cesqt, and establish that cesq is a proper subspace of cesqt whenever t ∈ [0, 1). Finally, we explore various geometric properties of these spaces, including the characterization of extreme points of the unit ball, strict convexity, (nearly) uniform convexity, (nearly) uniform smoothness, and property (β).